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

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

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);

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}]

Prog

(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);

Crossrefs

Cf. A057718.

Sequence in context: A091686 A091691 A024103 * A092661 A196071 A304303

Adjacent sequences: A187635 A187636 A187637 * A187639 A187640 A187641

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 12 2011

Status

approved