A187638 A convolution of binomial coefficients.
1, 1, 1, 8, 77, 824, 9448, 113728, 1419349, 18215576, 239034104, 3194168768, 43331889928, 595375647424, 8270102936896, 115966112244224, 1639592525879333, 23350785267093848, 334714889445506584, 4825695011761675456, 69936530485677013528
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Crossrefs
Cf. A057718.
Programs
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Maple
seq((-1/8)*add(binomial(2*k,k)^2/(2*k-1)*binomial(2*(n-k), n-k)^2/(2*(n-k)-1), k=0..n),n=1..12);
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Mathematica
Table[-1/8 Sum[Binomial[2k,k]^2/(2k-1) Binomial[2n-2k,n-k]^2/(2n-2k-1), {k, 0, n}], {n, 1, 20}] -
Maxima
makelist((-1/8)*sum(binomial(2*k,k)^2/(2*k-1)*binomial(2*(n-k),n-k)^2/(2*(n-k)-1),k,0,n),n,1,12);
Formula
G.f.: 9/8 - (1/8)*(K(16*x)/Pi)^2, where K(x) is the elliptic integral of the first kind (as defined in Mathematica).
a(n) = (1/8)*sum(C(2k,k)^2/(2k-1)*C(2n-2k,n-k)^2/(2n-2k-1), k=0..n) for n >= 1.
Recurrence: n^3*a(n) = 8*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 256*(n-3)*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Apr 06 2014
a(n) ~ 4^(2*n-1)/(Pi^2*n^2). - Vaclav Kotesovec, Apr 06 2014