A276537 Alternating binomial sums of the cubes of the central binomial coefficients.
1, 7, 201, 7375, 312265, 14365887, 697859169, 35226348087, 1829569294665, 97138289500735, 5248514415816721, 287657066913117447, 15953440327189548001, 893653778439275931175, 50488236061157830951545, 2873526763346873838886815
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..558
- Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind
- The Wolfram Functions Site, Complete Elliptic Integrals, 2016.
Crossrefs
Programs
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Magma
[&+[(-1)^(n-k)*Binomial(n,k)*Binomial(2*k,k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 03 2016
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Mathematica
Table[Sum[Binomial[n,k]Binomial[2k,k]^3(-1)^(n-k),{k,0,n}],{n,0,100}] -
Maxima
makelist(sum(binomial(n,k)*binomial(2*k,k)^3*(-1)^(n-k),k,0,n),n,0,12);
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*binomial(2*k,k)^3.
Recurrence: (n^3+12*n^2+48*n+64)*a(n+4)-(60*n^3+630*n^2+2204*n+2569)*a(n+3)-(186*n^3+1674*n^2+5037*n+5067)*a(n+2)-94*(2*n^3+15*n^2+37*n+30)*a(n+1)-63*(n^3+6*n^2+11*n+6)*a(n)=0.
G.f.: (4/Pi^2)*K(1/2-1/2*sqrt((1-63*t)/(1+t)))^2/(1+t), where K(x) is the complete elliptic integral of the first kind (defined as in MathWorld or in The Wolfram Functions Site).
a(n) ~ 3^(2*n+3) * 7^(n+3/2) / (512 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 16 2016
a(n) = (-1)^n*4F3(1/2,1/2,1/2,-n; 1,1,1; 64). - Ilya Gutkovskiy, Nov 25 2016