%I #39 Dec 08 2020 04:01:03

%S 1,-1,-3,-29,-499,-13101,-486131,-24266797,-1571357619,-128264296301,

%T -12894743113075,-1566235727656365,-226180775756251955,

%U -38308065207361046509,-7521255169156107737331,-1694604321825062440852013,-434302821056087233474158259

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

%H T. D. Noe, <a href="/A113871/b113871.txt">Table of n, a(n) for n = 0..100</a>

%H Marcelo Aguiar and Swapneel Mahajan, <a href="http://pi.math.cornell.edu/~maguiar/hadamard.pdf">On The Hadamard product Of Hopf monoids</a>

%H 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 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 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 a(0) = 1, a(n) = -Sum_{k=0..n-1} a(k) * ((n-k)!)^2. - _Daniel Suteu_, Feb 23 2018

%t nn = 20; CoefficientList[Series[1/Sum[(k!)^2 x^k, {k, 0, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jan 03 2013 *)

%o (Sage)

%o h = 1/(1+x*hypergeometric((1,2,2),(),x))

%o taylor(h,x,0,16).list() # _Peter Luschny_, Jul 28 2015

%o (Sage)

%o def A113871_list(len):

%o R, C = [1], [1]+[0]*(len-1)

%o for n in (1..len-1):

%o for k in range(n,-1,-1):

%o C[k] = C[k-1] * k^2

%o C[0] = -sum(C[k] for k in (1..n))

%o R.append(C[0])

%o return R

%o print(A113871_list(17)) # _Peter Luschny_, Jul 30 2015

%Y Cf. A003319, A051296, A113869, A114038, A316862.

%K sign

%O 0,3

%A _N. J. A. Sloane_, Jan 26 2006