A372723 Triangle read by rows: Column k has e.g.f. t^k / ((1 - t)^(k + 1) * exp(t)).

1, 0, 1, 1, 2, 2, 2, 9, 12, 6, 9, 44, 84, 72, 24, 44, 265, 640, 780, 480, 120, 265, 1854, 5430, 8520, 7560, 3600, 720, 1854, 14833, 50988, 97650, 112560, 78120, 30240, 5040, 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320

Offset: 0

Author

Peter Luschny, May 21 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]     1;
[1]     0,      1;
[2]     1,      2,      2;
[3]     2,      9,     12,       6;
[4]     9,     44,     84,      72,      24;
[5]    44,    265,    640,     780,     480,     120;
[6]   265,   1854,   5430,    8520,    7560,    3600,    720;
[7]  1854,  14833,  50988,   97650,  112560,   78120,  30240,   5040;
[8] 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320;
		

Crossrefs

Cf. A000166 (column 0), A000142 (main diagonal), A062119 (subdiagonal), A000354 (row sums), A033999 (alternating row sums), A372716 (central terms).

Programs

Maple

MAX := 14; gf := k -> t^k / ((1 - t)^(k + 1) * exp(t)):
ser := k -> series(gf(k), t, MAX):
col := k -> local n; seq(n!*coeff(series(ser(k), t, MAX-1), t, n), n = 0..MAX-2):
T := (n, k) -> col(k)[n+1]:
seq(lprint(seq(T(n, k), k = 0..n)), n = 0..8);