cp's OEIS Frontend

A187540 Binomial partial sums of the central Lah numbers.

%I A187540 #24 Feb 07 2017 21:35:32
%S A187540 1,3,41,1315,63825,4116611,331127353,31915763811,3585520583585,
%T A187540 460054836028675,66377105303195721,10637410917472061603,
%U A187540 1874707445757653437681,360356280811211873453955,75028021167256736753934425
%N A187540 Binomial partial sums of the central Lah numbers.
%H A187540 Vincenzo Librandi, <a href="/A187540/b187540.txt">Table of n, a(n) for n = 0..200</a>
%F A187540 Formula: a(n) = 1+sum(binomial(n,k)binomial(2k-1,k-1)(2k)!/k!,k=0..n).
%F A187540 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) )
%F A187540 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 A187540 a(n) ~ 16^n*n^(n-1/2)*exp(1/16-n)/sqrt(2*Pi). - _Vaclav Kotesovec_, Aug 09 2013
%p A187540 seq(1+add(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k=1..n), n=0..20);
%t A187540 Table[1 + Sum[Binomial[n, k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]
%o A187540 (Maxima) makelist(1+sum(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k,1,n), n,0,12);
%o A187540 (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
%Y A187540 Cf. A187536, A008297, A111596, A187538, A187539, A187542, A187543, A187544, A187545, A187546, A187547, A187548.
%K A187540 nonn,easy,nice
%O A187540 0,2
%A A187540 _Emanuele Munarini_, Mar 11 2011