A059370 -id:A059370 - cp's OEIS frontend

A059373 Second diagonal of triangle in A059370.

1, -4, 8, -16, 12, -96, -480, -4672, -45520, -493120, -5798912, -73668608, -1005335552, -14671085568, -228051746304, -3762955404288, -65707303602432, -1210821292674048, -23487031074109440, -478463919131627520, -10214440549929047040, -228069193578011566080

Offset: 2

N. J. A. Sloane, Jan 28 2001

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

Cf. A059370.

series(RootOf(T*hypergeom([1,2],[],T)-x,T)^2,x=0,21); # Mark van Hoeij, Apr 20 2013

nmax = 23; 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}]; A059370 = Reverse /@ Inverse[tnk] // DeleteCases[#, 0, 2] & ; Table[A059370[[n, n - 1]], {n, 2, nmax}] (* Jean-François Alcover, Jun 14 2013 *)

N = 66;  x = 'x + O('x^N);
tf = sum(n=1,N, n!*x^n );
gf=serreverse(%)^2;
v = Vec(gf)
/* Joerg Arndt, Apr 20 2013 */

G.f. A(x) is (R(x))^2 where R(x) is the series reversion of x*hypergeom([1,2],[],x) = sum(n>=1, n!*x^n), see Comtet. - Mark van Hoeij, Apr 20 2013

Added more terms, Mark van Hoeij and Joerg Arndt, Apr 20 2013

A059369 Triangle of numbers T(n,k) = T(n-1,k-1) + ((n+k-1)/k)*T(n-1,k), n >= 1, 1 <= k <= n, with T(n,1) = n!, T(n,n) = 1; read from right to left.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 24, 1, 8, 30, 72, 120, 1, 10, 48, 152, 372, 720, 1, 12, 70, 272, 828, 2208, 5040, 1, 14, 96, 440, 1576, 4968, 14976, 40320, 1, 16, 126, 664, 2720, 9696, 33192, 115200, 362880, 1, 18, 160, 952, 4380, 17312, 64704, 247968, 996480

Offset: 1

N. J. A. Sloane, Jan 28 2001

Another version of triangle in A090238. - Philippe Deléham, Jun 14 2007

			When read from left to right the rows {T(n,k), 1 <= k <= n} for n=1,2,3,... are 1; 2,1; 6,4,1; 24,16,6,1; ...

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

Cf. A059370, A059371.

nmax = 10; t[n_, k_] := Sum[(m+1)!*t[n-m-1, k-1], {m, 0, n-k}]; t[n_, 1] = n!; t[n_, n_] = 1; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, n, 1, -1}]] (* Jean-François Alcover, Nov 14 2011 *)

G.f. for k-th diagonal: (Sum_{i >= 1} i!*t^i)^k = Sum_{n >= k} T(n, k)*t^n.

T(n,k) = n! if k=1, 1 if k=n, Sum_{m=0..n-k} (m+1)!*T(n-m-1,k-1) otherwise. - Vladimir Kruchinin, Aug 18 2010

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 28 2001

First diagonal of triangle in A059370.

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

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

Cf. A000142, A059370.

# 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

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 *)

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

More terms from Vladeta Jovovic, Mar 05 2001

Definition refined by Georg Fischer, May 01 2021

cp's OEIS Frontend

A059373 Second diagonal of triangle in A059370.

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A059369 Triangle of numbers T(n,k) = T(n-1,k-1) + ((n+k-1)/k)*T(n-1,k), n >= 1, 1 <= k <= n, with T(n,1) = n!, T(n,n) = 1; read from right to left.

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

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Maple

Mathematica

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