A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).

1, -1, -1, -3, -13, -71, -461, -3447, -29093, -273343, -2829325, -31998903, -392743957, -5201061455, -73943424413, -1123596277863, -18176728317413, -311951144828863, -5661698774848621, -108355864447215063

Offset

0,4

Comments

Essentially the same as A003319, which is the main entry for these numbers. - N. J. A. Sloane, Jun 11 2013

References

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 40.

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Formula

a(n) = - Sum_{i=0..n-1} a(i)*(n-i)! for n > 0 with a(0) = 1 - Vladimir Kruchinin, Oct 10 2024

From Sergei N. Gladkovskii, Jun 24 2012, Oct 15 2012, Nov 18 2012, Dec 26 2012, Apr 25 2013, May 29 2013, Aug 08 2013, Nov 19 2013: (Start) Continued fractions:

G.f.: 1 - x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - (k+2)*x/Q(k+1)).

G.f.: U(0) where U(k) = 1 - x*(k+1)/(1 - x*(k+1)/U(k+1)).

G.f.: 1/G(0) where G(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))).

G.f.: A(x) = 1 - x/G(0) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).

G.f.: x*Q(0), where Q(k) = 1/x - 1 - 2*k - (k+1)^2/Q(k+1).

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).

G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )).

G.f.: conjecture: Q(0), where Q(k) = 1 + k*x - (k+1)*x/Q(k+1). (End)

a(n) ~ -n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10 - ...). - Vaclav Kotesovec, Dec 08 2020

Mathematica

CoefficientList[Series[1/(Sum[k!*x^k, {k, 0, 25}]), {x, 0, 20}], x] (* G. C. Greubel, Jun 30 2016 *)

Prog

(Maxima) a(n) := if n=0 then 1 else -sum( a(i)*(n-i)!, i, 0, n-1); \\ Vladimir Kruchinin, Oct 10 2024
(Sage)
def A167894_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * k
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A167894_list(20)) # Peter Luschny, Feb 19 2016
(Sage) m=20; (1/sum(factorial(k)*x^k for k in range(m+1))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019
(PARI) m=20; my(x='x+O('x^m)); Vec(1/sum(k=0, m+1, k!*x^k)) \\ G. C. Greubel, Feb 07 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&+[Factorial(k)*x^k: k in [0..m+1]]) )); // G. C. Greubel, Feb 07 2019

Crossrefs

Cf. A003319, A158882.

Sequence in context: A122455 A126390 A272428 * A158882 A233824 A003319

Adjacent sequences: A167891 A167892 A167893 * A167895 A167896 A167897

Keyword

sign

Author

Philippe Deléham, Nov 15 2009

Status

approved