cp's OEIS Frontend

A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.

%I A384494 #11 Jun 22 2025 00:16:04
%S A384494 1,2,-2,6,-4,3,24,-12,6,-4,120,-48,18,-8,5,720,-240,72,-24,10,-6,5040,
%T A384494 -1440,360,-96,30,-12,7,40320,-10080,2160,-480,120,-36,14,-8,362880,
%U A384494 -80640,15120,-2880,600,-144,42,-16,9,3628800,-725760,120960,-20160,3600,-720,168,-48,18,-10
%N A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.
%C A384494 This triangle, written as (infinite) square matrix MT with vanishing upper diagonals 0, together with the Riordan triangle A104698, written also as such a square matrix MR, appears in the double sum formula for the number of certain restricted permutations given in A086852(n), as diagonal sequence A086852(n+2) = (2*MR*MT^t)_{n,n}, for n >=0, where t indicates matrix transpositon.
%F A384494 T(n, k) = (-1)^k*(k+1)*(n+1-k)!, for n >= 0 and k = 0, 1, ..., n.
%F A384494 O.g.f. of row polynomials P(n, y) := Sum_{k=0..n} T(n, k) y^k: G(x, y) = ((N(x) - 1)/x) * (1/(1 + y*x)^2), with N(x) = hypergeometric([1,1], [], x), the o.g.f. of {n!}_{n>=0} (see A000142).
%e A384494 The triangle T begins:
%e A384494   n\k        0        1       2       3     4     5   6   7  8    9 ...
%e A384494   ---------------------------------------------------------------------
%e A384494   0:         1
%e A384494   1:         2       -2
%e A384494   2:         6       -4       3
%e A384494   3:        24      -12       6      -4
%e A384494   4:       120      -48      18      -8     5
%e A384494   5:       720     -240      72     -24    10    -6
%e A384494   6:      5040    -1440     360     -96    30   -12   7
%e A384494   7:     40320   -10080    2160    -480    12   -36  14  -8
%e A384494   8:    362880   -80640   15120   -2880   600  -144  42 -16  9
%e A384494   9:   3628800  -725760  120960  -20160  3600  -720 168 -48 18 -10
%e A384494   ...
%t A384494 Table[(-1)^k * (k+1) * (n+1-k)!, {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 31 2025 *)
%Y A384494 Cf. A104698, A086852, A086852.
%Y A384494 Column sequences: A000142(n+1), -A052849, A052560(n-1), -A052578(n-2), A052648(n-3), -A298881(n-4), A062098(n-5), -A159038(n-6), ...
%K A384494 sign,tabl,easy
%O A384494 0,2
%A A384494 _Wolfdieter Lang_, May 31 2025