A187540 Binomial partial sums of the central Lah numbers.

1, 3, 41, 1315, 63825, 4116611, 331127353, 31915763811, 3585520583585, 460054836028675, 66377105303195721, 10637410917472061603, 1874707445757653437681, 360356280811211873453955, 75028021167256736753934425

Offset

0,2

Links

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

Formula

Formula: a(n) = 1+sum(binomial(n,k)binomial(2k-1,k-1)(2k)!/k!,k=0..n).

Recurrence: for n>=3, a(n) = 1/n*(-2 +(32 - 48*n + 16*n^2)*a(n-3) + (-31 + 63*n - 32*n^2)*a(n-2) + (3 - 14*n + 16*n^2)*a(n-1) )

E.g.f.: exp(x) (1/2 + 1/Pi K(16x) ), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).

a(n) ~ 16^n*n^(n-1/2)*exp(1/16-n)/sqrt(2*Pi). - Vaclav Kotesovec, Aug 09 2013

Maple

seq(1+add(binomial(n, k)*binomial(2*k-1, k-1)*(2*k)!/k!, k=1..n), n=0..20);

Mathematica

Table[1 + Sum[Binomial[n, k]Binomial[2k-1, k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]

Prog

(Maxima) makelist(1+sum(binomial(n, k)*binomial(2*k-1, k-1)*(2*k)!/k!, k, 1, n), n, 0, 12);
(PARI) a(n) = 1+sum(k=0, n, binomial(n, k)*binomial(2*k-1, k-1)*(2*k)!/k!) \\ Charles R Greathouse IV, Feb 07 2017

Crossrefs

Cf. A187536, A008297, A111596, A187538, A187539, A187542, A187543, A187544, A187545, A187546, A187547, A187548.

Sequence in context: A012035 A012016 A207993 * A012104 A012147 A012011

Adjacent sequences: A187537 A187538 A187539 * A187541 A187542 A187543

Keyword

nonn,easy,nice

Author

Emanuele Munarini, Mar 11 2011

Status

approved