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

0,3

Links

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

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

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}]

Prog

(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);

Crossrefs

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

Sequence in context: A009977 A371760 A293693 * A130835 A262101 A077420

Adjacent sequences: A187536 A187537 A187538 * A187540 A187541 A187542

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 11 2011

Status

approved