A187539 Alternated binomial partial sums of central Lah numbers (A187535).

1, 1, 33, 1097, 54209, 3527889, 285356449, 27608615257, 3110179582593, 399896866564001, 57791843384031521, 9273757516482276201, 1636151050649025202753, 314786007405793614831217, 65590496972310741712688289, 14714600180590751334321307769

Offset: 0

Author

Emanuele Munarini, Mar 11 2011

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187540, A187542, A187543, A187544, A187545, A187546, A187547, A187548.

Programs

Maple

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

Mathematica

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

Maxima

makelist((-1)^n+sum((-1)^(n-k)*binomial(n,k)*binomial(2*k-1,k-1) *(2*k)!/k!, k,1,n), n,0,12);

Formula

a(n) = 1+sum((-1)^(n-k)*C(n,k)*C(2k-1,k-1)*(2k)!/k!, k=0..n).

Recurrence: n>=3, a(n) = (2*(-1)^n + (32 - 48*n + 16*n^2)*a(n-3) + (33 - 65*n + 32*n^2)*a(n-2) + (5 - 18*n + 16*n^2)*a(n-1))/n

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)/(sqrt(2*Pi)*exp(n+1/16)). - Vaclav Kotesovec, Aug 10 2013