A316862 -id:A316862 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 26 2006

sign

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.

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

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

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

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

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

A306629 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} (j!)^k * x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, 0, 1, -1, -3, -3, 0, 1, -1, -7, -29, -13, 0, 1, -1, -15, -201, -499, -71, 0, 1, -1, -31, -1265, -13351, -13101, -461, 0, 1, -1, -63, -7713, -328975, -1697705, -486131, -3447, 0, 1, -1, -127, -46529, -7946143, -206659569, -369575303, -24266797, -29093, 0

Offset: 0

Seiichi Manyama, Mar 02 2019

			Square array begins:
    1,    1,       1,          1,             1, ...
   -1,   -1,      -1,         -1,            -1, ...
    0,   -1,      -3,         -7,           -15, ...
    0,   -3,     -29,       -201,         -1265, ...
    0,  -13,    -499,     -13351,       -328975, ...
    0,  -71,  -13101,   -1697705,    -206659569, ...
    0, -461, -486131, -369575303, -268312660751, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns 1-3 give A167894, A113871, A316862.
Rows 0-2 give A000012, (-1)*A000012, (-1)*A000225.
Main diagonal gives A306630.

A(0,k) = 1 and A(n,k) = -Sum_{j=1..n} (j!)^k * A(n-j,k) for n > 0.

cp's OEIS Frontend

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

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