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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Crossrefs
Programs
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Maple
seq(1+add(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k=1..n), n=0..20);
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Mathematica
Table[1 + Sum[Binomial[n, k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}] -
Maxima
makelist(1+sum(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k,1,n), n,0,12);
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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
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