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
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
Programs
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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
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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 *)
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
Extensions
More terms from Vladeta Jovovic, Mar 05 2001
Definition refined by Georg Fischer, May 01 2021
Comments