A059372 Revert transform of factorials n! (n >= 1).

1, -2, 2, -4, -4, -48, -336, -2928, -28144, -298528, -3454432, -43286528, -583835648, -8433987584, -129941213184, -2127349165824, -36889047574272, -675548628690432, -13030733384956416, -264111424634864640

Offset

1,2

Comments

First diagonal of triangle in A059370.

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..400 (first 100 terms from T. D. Noe)

M. H. Albert, M. D. Atkinson and M. Klazar, The Enumeration of Simple Permutations, J. Integer Seqs., Vol. 6, 2003.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Blockwise simple permutations, arXiv:2303.13115 [math.CO], 2023.

Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.

Index entries for reversions of series

Formula

a(n) ~ -exp(-2) * n! * (1 - 4/n + 2/n^2 - 34/(3*n^3) - 296/(3*n^4) - 4818/(5*n^5) - 508532/(45*n^6)). - Vaclav Kotesovec, Aug 04 2015

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k! * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020

Maple

# From Transforms, see the footer of the page.
REVERT([seq(k!, k=1..20)]); # Peter Luschny, May 01 2021
# Using function CompInv from A357588.
CompInv(10, n -> factorial(n)); # Peter Luschny, Oct 09 2022

Mathematica

nmax = 20; t[n_, k_] := t[n, k] = Sum[(m + 1)!*t[n - m - 1, k - 1], {m, 0, n - k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; Inverse[tnk][[All, 1]] (* Jean-François Alcover, Jul 13 2016 *)

Crossrefs

Cf. A000142, A059370.

Sequence in context: A025557 A285909 A322253 * A161422 A049261 A135018

Adjacent sequences: A059369 A059370 A059371 * A059373 A059374 A059375

Keyword

sign,easy

Author

N. J. A. Sloane, Jan 28 2001

Extensions

More terms from Vladeta Jovovic, Mar 05 2001

Definition refined by Georg Fischer, May 01 2021

Status

approved