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A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

%I A372311 #10 Apr 27 2024 03:32:12
%S A372311 1,1,1,1,6,8,1,21,108,162,1,60,800,3840,6144,1,155,4500,48750,225000,
%T A372311 375000,1,378,21672,453600,4354560,19595520,33592320,1,889,94668,
%U A372311 3500658,60505200,536479440,2371803840,4150656720
%N A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).
%e A372311 Triangle begins:
%e A372311   [0] 1;
%e A372311   [1] 1,   1;
%e A372311   [2] 1,   6,     8;
%e A372311   [3] 1,  21,   108,     162;
%e A372311   [4] 1,  60,   800,    3840,     6144;
%e A372311   [5] 1, 155,  4500,   48750,   225000,    375000;
%e A372311   [6] 1, 378, 21672,  453600,  4354560,  19595520,   33592320;
%e A372311   [7] 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720;
%p A372311 S := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
%p A372311 row := n -> local k; seq(S(n, k) * n^k, k = 0..n):
%p A372311 seq(row(n), n = 0..8);
%o A372311 (SageMath)
%o A372311 def A372311_row(n) :
%o A372311     x = polygen(ZZ, 'x')
%o A372311     A = []
%o A372311     for m in range(0, n + 1, 1) :
%o A372311         A.append((-x)^m)
%o A372311         for j in range(m, 0, -1):
%o A372311             A[j - 1] = j * (A[j - 1] - A[j])
%o A372311     return [n^k*c for k, c in enumerate(A[0])]
%o A372311 for n in (0..7) : print(A372311_row(n))
%Y A372311 Cf. A061711 (main diagonal), A066524 (column 1), A372312 (row sums).
%Y A372311 Cf. A163626, A173018 (eulerian1).
%K A372311 nonn,tabl
%O A372311 0,5
%A A372311 _Peter Luschny_, Apr 26 2024