#### #### #### Order 2 #### #### #### SS[1]:=[ 1/ (x*y*z + x*z - x - y - z + 1), (x^3+11*x^2-x)*Dx^2+(3*x^2+22*x-1)*Dx+x+3]: # hypergeom([1/12, 5/12],[1], 1728*x^5*(1-11*x-x^2)/(1-12*x+14*x^2+12*x^3+x^4)^3) / (1-12*x+14*x^2+12*x^3+x^4)^(1/4) # HeunG((11-5*5^(1/2))/(11+5*5^(1/2)),(1476+660*5^(1/2))/(11+5*5^(1/2))^3,1,1,1,1,-2/(11+5*5^(1/2))*x) # Apery numbers: a(n) = sum_{k=0..n} C(n,k)^2 * C(n+k,k) # http://oeis.org/A005258 SS[2] :=[ 1/ (x*z - x - y - z + 1), (16*x^2-x)*Dx^2+(32*x-1)*Dx+4 ]: # Diag( 1/ ((-x - y + 1) * (-t - z + 1)), 6): # 2*EllipticK(4*x^(1/2))/Pi # Binomial(2n,n)^2. # http://oeis.org/A002894 # hypergeom([1/12, 5/12],[1],1728*x^4*(1-16*x)/(1-16*x+16*x^2)^3)/(1-16*x+16*x^2)^(1/4) SS[3] := [1/ (-x*y*z - x*z - y*z - x - y + 1), x*(x-1)*(x+1)*(x^2-18*x+1)*Dx^2 + (3*x^4-34*x^3-38*x^2+38*x-1)*Dx + x^3-3*x^2-19*x+5]: # a(n) = Sum_{k=0..n} C(2*k, k)^2 * C(n+k, n-k). # http://oeis.org/A243945 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^4*(x^2-18*x+1)*(x-1)^2/(x^4-20*x^3+54*x^2-20*x+1)^3)/(x^4-20*x^3+54*x^2-20*x+1)^(1/4): SS[4]:= [ 1/ (-x*y - x*z - y*z - x - y + 1), x*(x+4)*(x+1)*(27*x^2+34*x-2)*Dx^2+(81*x^4+554*x^3+764*x^2+256*x-8)*Dx+24*x^3+184*x^2+192*x+32]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12], [1], 1728*x^4*(x+1)^2*(27*x^2+34*x-2)/(-1+16*x+8*x^2)^3)/(1-16*x-8*x^2)^(1/4): SS[5] := [ 1 / (x*y*z - x*y + x*z - x - y - z + 1), x*(5*x+8)*(x^3+5*x^2+39*x-2)*Dx^2 +(15*x^4+82*x^3+315*x^2+624*x-16)*Dx + 5*x^3+21*x^2+80]: # http://oeis.org/A112019 # Sum(C(n,k)*C(n+k,k)^2,k=0..n) # G.f.: y(x) = hypergeom([1/12, 5/12], [1], -1728*(x^3+5*x^2+39*x-2)*x^4 / (x^4+4*x^3+30*x^2-20*x+1)^3 ) / (x^4+4*x^3+30*x^2-20*x+1)^(1/4): SS[6]:=[ 1/ (-x*y - y*z - x - y - z + 1), x*(4*x+3)*(16*x^2+44*x-1)*Dx^2 +(128*x^3+320*x^2+264*x-3)*Dx+16*x^2+12*x+30]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3)/(1-40*x+16*x^2)^(1/4): SS[7] := [ 1/ (-x*y*z - x*y - y*z - x - y - z + 1), x*(x-2)*(5*x+2)*(3*x^2-47*x+1)*Dx^2 +(45*x^4-506*x^3+157*x^2+380*x-4)*Dx+15*x^3-63*x^2-136*x+44]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12], [1],1728*(3*x^2-47*x+1)*(x-2)^2*x^3/(x^4-20*x^3+78*x^2-44*x+1)^3)/(x^4-20*x^3+78*x^2-44*x+1)^(1/4): SS[8] := [1 / (-x*y - x - y - z + 1), x*(2*x+3)*(16*x^2-71*x+2)*Dx^2+2*(32*x^3+x^2-213*x+3)*Dx+8*x^2+48*x-48] : # not in oeis # G.f.: y(x) = hypergeom([1/12,5/12],[1],1728*x^3*(16*x^2-71*x+2)/(16*x^2-32*x+1)^3)/(16*x^2-32*x+1)^(1/4): SS[9] := [ 1 / (-x*y*z - x*y - x - y - z + 1), x*(5*x^2-12*x-6)*(x^4-13*x^3+77*x^2-78*x+2)*Dx^2+(15*x^6-178*x^5+823*x^4-1536*x^3-460*x^2+936*x-12)*Dx + 5*x^5-39*x^4+48*x^3 +48*x^2-420*x+108] : # not in oeis # G.f.: y(x) = hypergeom([1/12,5/12],[1],1728*x^3*(x^4-13*x^3+77*x^2-78*x+2)/(x^4-12*x^3+62*x^2-36*x+1)^3)/(x^4-12*x^3+62*x^2-36*x+1)^(1/4): SS[10] := [ 1/ (-x - y - z + 1) , x*(27*x-1)*Dx^2+(54*x-1)*Dx + 6 ] : # De Bruijn's S(3,n): (3n)!/(n!)^3 # http://oeis.org/A006480 # G.f.: y(x) = hypergeom([1/3, 2/3], [1], 27*x): # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^3*(1-27*x)/(1-24*x)^3)/(1-24*x)^(1/4): SS[11] := [ 1/ (-x*y*z - x - y - z + 1) , (2*x+1)*(x^3-3*x^2+30*x-1)*x*Dx^2 +(6*x^4-8*x^3+51*x^2+60*x-1)*Dx + (x-1)*(2*x^2+2*x-7) ] : # G.f.: y(x) = hypergeom([1/3, 2/3],[1],27*x/(1-x)^3)/(1-x) # Sum_{k=0..n} C(n,k) * C(n+k,k) * C(n+2*k,k). # http://oeis.org/A081798 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^3*(1-30*x+3*x^2-x^3)/(1-28*x+30*x^2-4*x^3+x^4)^3)/(1-28*x+30*x^2-4*x^3+x^4)^(1/4): SS[12] := [ 1/ (-x*y - x*z - y*z - x - y - z + 1), x*(x+2)*(x+1)*(27*x^2+54*x-1)*Dx^2+(81*x^4+324*x^3+431*x^2+214*x-2)*Dx+24*x^3+72*x^2+72*x+24 ]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12], [1], 1728*x^3*(x+2)^3*(27*x^2+54*x-1)/(-1+48*x+24*x^2)^3)/(1-48*x-24*x^2)^(1/4): SS[13] := [ 1/ (-x*y*z - x*y - x*z - y*z - x - y - z + 1) , (2*x+1)*(x^3-3*x^2+57*x-1)*x*Dx^2+(6*x^4-8*x^3+105*x^2+114*x-1)*Dx+(x-1)*(2*x^2+2*x-13) ]: # G.f.: y(x) = hypergeom([1/3,2/3],[1], 54*x/(1-x)^3)/(1-x) # http://oeis.org/A126086 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],13824*x^3*(1-57*x+3*x^2-x^3)/(1-52*x+54*x^2-4*x^3+x^4)^3)/(1-52*x+54*x^2-4*x^3+x^4)^(1/4): SS[14] := [ 1/ (-x*y - x*z - y*z - z + 1) , x*(2*x+5)*(27*x^3+20*x^2+7*x-1)*Dx^2+(162*x^4+620*x^3+314*x^2+70*x-5)*Dx+48*x^3+220*x^2+80*x+10 ] : # not in oeis.org # G.f.: hypergeom([1/12, 5/12], [1], 1728*x^5*(27*x^3+20*x^2+7*x-1)/(8*x^2+8*x-1)^3)/(1-8*x-8*x^2)^(1/4): SS[15]:=[ 1/(-x * y * z - x * y - x * z - y * z + 1) , (x+2)*(x^3+24*x^2+3*x-1)*x*Dx^2 +(3*x^4+56*x^3+147*x^2+12*x-2)*Dx +x^3+9*x^2+42*x+2 ]: # hypergeom([1/3, 2/3], [1], 27*x^2/(1-x)^3)/(1-x) # http://oeis.org/A208425 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^6*(1-3*x-24*x^2-x^3)/(1-4*x-18*x^2+20*x^3+x^4)^3)/(1-4*x-18*x^2+20*x^3+x^4)^(1/4): ################# SS[16] := [ 1/ (x*y + x*z - y*z - x - y - z + 1), 8 - 24*x + Dx*(-2 + 58*x - 81*x^2) + Dx^2*(-2*x + 29*x^2 - 27*x^3)]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],3456*x^5*(1-31/2*x+28*x^2-27/2*x^3)/(1-16*x+40*x^2)^3)/(1-16*x+40*x^2)^(1/4): SS[17] := [1/ (-x*y*z + x*y - x*z - y*z - x - y + 1), x*(2*x+5)*(2*x-1)*(x^2-47*x+3)*Dx^2+(12*x^4-340*x^3-1319*x^2+530*x-15)*Dx+4*x^3-24*x^2-445*x+75]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^5*(x^2-47*x+3)*(-1+2*x)^2/(1-20*x+78*x^2-44*x^3+x^4)^3)/(1-20*x+78*x^2-44*x^3+x^4)^(1/4): SS[18] := [ 1/ (x*y*z - x*z - y*z - x - y + 1), x*(x-1)*(x+1)*(x^2-14*x+1)*Dx^2 +(3*x^4-30*x^3+26*x^2+26*x-1)*Dx + x^3-5*x^2+13*x+3]: # http://oeis.org/A245925 # The g.f.s formed from a(2*n)^(1/2) and (-a(2*n+1)/3)^(1/2) are: # A245926: sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) ): # A245927: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ). # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^4*(x^2-14*x+1)*(x+1)^2/(1-12*x-10*x^2-12*x^3+x^4)^3)/(1-12*x-10*x^2-12*x^3+x^4)^(1/4): SS[19] := [ 1/ (x*y*z - x*y - x*z - y*z - x - y + 1), x*(2*x+1)*(6*x^2+x-8)*(x^3-41*x^2-29*x+2)*Dx^2+(36*x^6-964*x^5-917*x^4+2394*x^3+2339*x^2+400*x-16)*Dx+12*x^5-104*x^4 +57*x^3+1067*x^2+640*x+48]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^4*(x^3-41*x^2-29*x+2)*(1+2*x)^2/(1-12*x-34*x^2-36*x^3+x^4)^3)/(1-12*x-34*x^2-36*x^3+x^4)^(1/4): SS[20] := [ 1/ (x*y*z - x*y + x*z - y*z - x - y - z + 1), x*(5*x+4)*(x^3+19*x^2+27*x-1)*Dx^2+(15*x^4+206*x^3+363*x^2+216*x-4)*Dx+5*x^3+33*x^2+36*x+28]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],13824*x^4*(1-27*x-19*x^2-x^3)/(1-28*x+54*x^2+20*x^3+x^4)^3)/(1-28*x+54*x^2+20*x^3+x^4)^(1/4): SS[21]:=[ 1/ (x*y - x*z - y*z - x - y - z + 1) , x*(x-2)*(11*x+12)*(27*x^2+92*x-3)*Dx^2 +(891*x^4+2132*x^3-689*x^2-4488*x+72)*Dx + 264*x^3+592*x^2+768*x-576 ]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],-1728*x^4*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3)/(1-32*x+88*x^2)^(1/4): SS[22] := [ 1/ (-x*y*z - x*y + x*z - y*z - x - y - z + 1), x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2+(3*x^4-66*x^3-70*x^2+70*x-1)*Dx+x^3-7*x^2-35*x+9 ]: # Diag(1/((-x*y - x - y + 1) * (-t * z - t - z + 1)), 6): # Squares of the central Delannoy numbers: a(n) = A001850(n)^2 # http://oeis.org/A243949 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4): SS[23]:=[ 1 / (-x*y + x*z - x - y - z + 1), x*(16*x^2-24*x+1)*Dx^2+(48*x^2-48*x+1)*Dx+12*x-6]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],6912*x^4*(1-24*x+16*x^2)/(1-24*x+48*x^2)^3)/(1-24*x+48*x^2)^(1/4): SS[24]:= [ 1/ (-x*y*z - x*y + x*z - x - y - z + 1), x*(8*x^2-5*x-24)*(2*x^4-9*x^3+205*x^2-167*x+6)*Dx^2+(48*x^6-184*x^5+1535*x^4-1186*x^3-13973*x^2+8016*x-144)*Dx +16*x^5-36*x^4-491*x^3+839*x^2-3840*x+1008 ] : # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^4*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3)/(1-28*x+78*x^2-4*x^3+x^4)^(1/4): SS[25] := [ 1/ (-x*y*z - x*y + x*z + y*z - x - y - z + 1) , x*(4*x-5)*(x^3+27*x^2-19*x+1)*Dx^2+(12*x^4+196*x^3-481*x^2+190*x-5)*Dx+4*x^3+18*x^2-95*x+25 ]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],13824*x^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4): SS[26] := [ 1 / (x*y*z - x*y - y*z - x - y - z + 1), x*(x+2)*(8*x^2-x-6)*(2*x^3-29*x^2-41*x+1)*Dx^2+(48*x^6-408*x^5-777*x^4+798*x^3+1847*x^2+972*x-12)*Dx +16*x^5-52*x^4+141*x^3+529*x^2+408*x+108 ]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^3*(2*x^3-29*x^2-41*x+1)*(x+2)^2/(1-36*x-34*x^2-12*x^3+x^4)^3)/(1-36*x-34*x^2-12*x^3+x^4)^(1/4): SS[27] := [1 / (x*y*z - x*y - x - y - z + 1), x*(5*x^2+4*x-6)*(x^4-3*x^3-27*x^2-64*x+2)*Dx^2+(15*x^6-14*x^5-201*x^4-144*x^3+220*x^2+768*x-12)*Dx +5*x^5+3*x^4+136*x^2+132*x+84 ]: # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^3*(2-64*x-27*x^2-3*x^3+x^4)/(1-28*x-18*x^2-4*x^3+x^4)^3)/(1-28*x-18*x^2-4*x^3+x^4)^(1/4): SS[28] := [ 1/ (x*y*z - x - y - z + 1) , (2*x-1)*(x^3+3*x^2-24*x+1)*x*Dx^2 +(6*x^4+8*x^3-57*x^2+48*x-1)*Dx +(x+1)*(2*x^2-2*x+5) ]: # G.f.: y(x) = hypergeom([1/3, 2/3],[1],27*x/(1+x)^3)/(1+x) # Number of effective multiple alignments of three equal-length sequences # http://oeis.org/A124435 # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^3*(1-24*x+3*x^2+x^3)/(1-20*x-18*x^2+4*x^3+x^4)^3)/(1-20*x-18*x^2+4*x^3+x^4)^(1/4): SS[29] := [ 1/ (x*y*z + x*y + x*z - y*z - x + y + z + 1) , x*(x-1)*(x^2-52*x+1)*(x+1)^2*Dx^2+(x+1)*(3*x^4-106*x^3+102*x^2+102*x-1)*Dx+x^4-12*x^3+32*x^2+68*x+11 ] : # not in oeis # G.f.: y(x) = hypergeom([1/12, 5/12],[1],13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2)/((x^2-46*x+1)*(x+1)^2)^(1/4): SS[30] := [ 1 / (x*y*z - x*y - x*z - y*z - x + 1) , x*(6*x^2-4*x-5)*(2*x^4-64*x^3-27*x^2-3*x+1)*Dx^2 +(36*x^6-800*x^5+556*x^4+1496*x^3+411*x^2+30*x-5)*Dx +12*x^5-100*x^4+256*x^3+540*x^2+105*x+5 ] : # not in oeis.org # G.f.: y(x) = hypergeom([1/12, 5/12],[1],1728*x^5*(1-3*x-27*x^2-64*x^3+2*x^4)/(1-4*x-18*x^2-28*x^3+x^4)^3)/(1-4*x-18*x^2-28*x^3+x^4)^(1/4):