cp's OEIS Frontend

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

%I A187539 #18 Aug 10 2013 13:50:17
%S A187539 1,1,33,1097,54209,3527889,285356449,27608615257,3110179582593,
%T A187539 399896866564001,57791843384031521,9273757516482276201,
%U A187539 1636151050649025202753,314786007405793614831217,65590496972310741712688289,14714600180590751334321307769
%N A187539 Alternated binomial partial sums of central Lah numbers (A187535).
%H A187539 Vincenzo Librandi, <a href="/A187539/b187539.txt">Table of n, a(n) for n = 0..200</a>
%F A187539 a(n) = 1+sum((-1)^(n-k)*C(n,k)*C(2k-1,k-1)*(2k)!/k!, k=0..n).
%F A187539 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
%F A187539 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).
%F A187539 a(n) ~ 16^n*n^(n-1/2)/(sqrt(2*Pi)*exp(n+1/16)). - _Vaclav Kotesovec_, Aug 10 2013
%p A187539 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);
%t A187539 Table[(-1)^n + Sum[(-1)^(n-k)Binomial[n,k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]
%o A187539 (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);
%Y A187539 Cf. A187536, A008297, A111596, A187536, A187538, A187540, A187542, A187543, A187544, A187545, A187546, A187547, A187548.
%K A187539 nonn,easy
%O A187539 0,3
%A A187539 _Emanuele Munarini_, Mar 11 2011