cp's OEIS Frontend

A113871 G.f.: 1/(Sum_{k>=0} (k!)^2 x^k).

%I A113871 #39 Dec 08 2020 04:01:03
%S A113871 1,-1,-3,-29,-499,-13101,-486131,-24266797,-1571357619,-128264296301,
%T A113871 -12894743113075,-1566235727656365,-226180775756251955,
%U A113871 -38308065207361046509,-7521255169156107737331,-1694604321825062440852013,-434302821056087233474158259
%N A113871 G.f.: 1/(Sum_{k>=0} (k!)^2 x^k).
%H A113871 T. D. Noe, <a href="/A113871/b113871.txt">Table of n, a(n) for n = 0..100</a>
%H A113871 Marcelo Aguiar and Swapneel Mahajan, <a href="http://pi.math.cornell.edu/~maguiar/hadamard.pdf">On The Hadamard product Of Hopf monoids</a>
%H A113871 John D. Dixon, <a href="https://doi.org/10.37236/1953">Asymptotics of Generating the Symmetric and Alternating Groups</a>, Electronic Journal of Combinatorics, 2005, vol 11(2), R56.
%F A113871 G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - (k+1)^2*x/((k+1)^2*x + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 17 2013
%F A113871 a(n) ~ -n!^2 * (1 - 2/n^2 - 5/n^4 - 10/n^5 - 67/n^6 - 332/n^7 - 2152/n^8 - 14946/n^9 - 115583/n^10). - _Vaclav Kotesovec_, Jul 28 2015
%F A113871 a(0) = 1, a(n) = -Sum_{k=0..n-1} a(k) * ((n-k)!)^2. - _Daniel Suteu_, Feb 23 2018
%t A113871 nn = 20; CoefficientList[Series[1/Sum[(k!)^2 x^k, {k, 0, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jan 03 2013 *)
%o A113871 (Sage)
%o A113871 h = 1/(1+x*hypergeometric((1,2,2),(),x))
%o A113871 taylor(h,x,0,16).list() # _Peter Luschny_, Jul 28 2015
%o A113871 (Sage)
%o A113871 def A113871_list(len):
%o A113871     R, C = [1], [1]+[0]*(len-1)
%o A113871     for n in (1..len-1):
%o A113871         for k in range(n,-1,-1):
%o A113871             C[k] = C[k-1] * k^2
%o A113871         C[0] = -sum(C[k] for k in (1..n))
%o A113871         R.append(C[0])
%o A113871     return R
%o A113871 print(A113871_list(17)) # _Peter Luschny_, Jul 30 2015
%Y A113871 Cf. A003319, A051296, A113869, A114038, A316862.
%K A113871 sign
%O A113871 0,3
%A A113871 _N. J. A. Sloane_, Jan 26 2006