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

1, -1, -3, -29, -499, -13101, -486131, -24266797, -1571357619, -128264296301, -12894743113075, -1566235727656365, -226180775756251955, -38308065207361046509, -7521255169156107737331, -1694604321825062440852013, -434302821056087233474158259

Offset: 0

Author

N. J. A. Sloane, Jan 26 2006

Keywords

• sign

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Marcelo Aguiar and Swapneel Mahajan, On The Hadamard product Of Hopf monoids
• John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, 2005, vol 11(2), R56.

Crossrefs

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

Programs

Mathematica

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

Sage

h = 1/(1+x*hypergeometric((1,2,2),(),x))
taylor(h,x,0,16).list() # Peter Luschny, Jul 28 2015

Sage

def A113871_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n,-1,-1):
            C[k] = C[k-1] * k^2
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A113871_list(17)) # Peter Luschny, Jul 30 2015

Formula

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

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

a(0) = 1, a(n) = -Sum_{k=0..n-1} a(k) * ((n-k)!)^2. - Daniel Suteu, Feb 23 2018