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%I #7 May 21 2024 10:20:05
%S 1,0,1,1,2,2,2,9,12,6,9,44,84,72,24,44,265,640,780,480,120,265,1854,
%T 5430,8520,7560,3600,720,1854,14833,50988,97650,112560,78120,30240,
%U 5040,14833,133496,526568,1189104,1681680,1525440,866880,282240,40320
%N Triangle read by rows: Column k has e.g.f. t^k / ((1 - t)^(k + 1) * exp(t)).
%e Triangle starts:
%e [0] 1;
%e [1] 0, 1;
%e [2] 1, 2, 2;
%e [3] 2, 9, 12, 6;
%e [4] 9, 44, 84, 72, 24;
%e [5] 44, 265, 640, 780, 480, 120;
%e [6] 265, 1854, 5430, 8520, 7560, 3600, 720;
%e [7] 1854, 14833, 50988, 97650, 112560, 78120, 30240, 5040;
%e [8] 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320;
%p MAX := 14; gf := k -> t^k / ((1 - t)^(k + 1) * exp(t)):
%p ser := k -> series(gf(k), t, MAX):
%p col := k -> local n; seq(n!*coeff(series(ser(k), t, MAX-1), t, n), n = 0..MAX-2):
%p T := (n, k) -> col(k)[n+1]:
%p seq(lprint(seq(T(n, k), k = 0..n)), n = 0..8);
%Y Cf. A000166 (column 0), A000142 (main diagonal), A062119 (subdiagonal), A000354 (row sums), A033999 (alternating row sums), A372716 (central terms).
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, May 21 2024