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

%I A372723 #7 May 21 2024 10:20:05
%S A372723 1,0,1,1,2,2,2,9,12,6,9,44,84,72,24,44,265,640,780,480,120,265,1854,
%T A372723 5430,8520,7560,3600,720,1854,14833,50988,97650,112560,78120,30240,
%U A372723 5040,14833,133496,526568,1189104,1681680,1525440,866880,282240,40320
%N A372723 Triangle read by rows: Column k has e.g.f. t^k / ((1 - t)^(k + 1) * exp(t)).
%e A372723 Triangle starts:
%e A372723 [0]     1;
%e A372723 [1]     0,      1;
%e A372723 [2]     1,      2,      2;
%e A372723 [3]     2,      9,     12,       6;
%e A372723 [4]     9,     44,     84,      72,      24;
%e A372723 [5]    44,    265,    640,     780,     480,     120;
%e A372723 [6]   265,   1854,   5430,    8520,    7560,    3600,    720;
%e A372723 [7]  1854,  14833,  50988,   97650,  112560,   78120,  30240,   5040;
%e A372723 [8] 14833, 133496, 526568, 1189104, 1681680, 1525440, 866880, 282240, 40320;
%p A372723 MAX := 14; gf := k -> t^k / ((1 - t)^(k + 1) * exp(t)):
%p A372723 ser := k -> series(gf(k), t, MAX):
%p A372723 col := k -> local n; seq(n!*coeff(series(ser(k), t, MAX-1), t, n), n = 0..MAX-2):
%p A372723 T := (n, k) -> col(k)[n+1]:
%p A372723 seq(lprint(seq(T(n, k), k = 0..n)), n = 0..8);
%Y A372723 Cf. A000166 (column 0), A000142 (main diagonal), A062119 (subdiagonal), A000354 (row sums), A033999 (alternating row sums), A372716 (central terms).
%K A372723 nonn,tabl
%O A372723 0,5
%A A372723 _Peter Luschny_, May 21 2024