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A362995 Triangle read by rows. T(n, k) = [x^k] lcm({i + 1 : 0 <= i <= n}) * (Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1)).

%I A362995 #18 Jan 15 2025 06:10:15
%S A362995 1,3,2,11,28,18,25,184,351,192,137,2608,11097,16128,7500,147,6816,
%T A362995 57591,166912,193750,77760,1089,118464,1865511,9588736,20843750,
%U A362995 20062080,7058940,2283,567936,16015401,136921088,495546875,858003840,704129265,220200960
%N A362995 Triangle read by rows. T(n, k) = [x^k] lcm({i + 1 : 0 <= i <= n}) * (Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1)).
%H A362995 Peter Luschny, <a href="https://math.stackexchange.com/questions/4706827/">An integer triangle for representing the Bernoulli numbers</a>, Mathematics Stack Exchange, May 2023.
%H A362995 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A362995 T(n, k) = lcm(1,2, ..., n+1) * A362996(n, k) / A362997(n, k).
%F A362995 Sum_{k=0..n} (-1)^k * T(n, k) = lcm(1,2, ..., n+1) * Bernoulli(n, 1) = A362994(n).
%e A362995 Triangle T(n, k) starts:
%e A362995 [0]    1;
%e A362995 [1]    3,      2;
%e A362995 [2]   11,     28,       18;
%e A362995 [3]   25,    184,      351,       192;
%e A362995 [4]  137,   2608,    11097,     16128,      7500;
%e A362995 [5]  147,   6816,    57591,    166912,    193750,     77760;
%e A362995 [6] 1089, 118464,  1865511,   9588736,  20843750,  20062080,   7058940;
%e A362995 [7] 2283, 567936, 16015401, 136921088, 495546875, 858003840, 704129265, 220200960;
%p A362995 R := (n, x) -> add(add(x^j*binomial(u, j)*(j + 1)^n, j = 0..u)/(u + 1), u=0..n):
%p A362995 CoeffList := p -> PolynomialTools:-CoefficientList(p, x):
%p A362995 poly := (n, x) -> ilcm(seq(i, i = 1..n+1)) * R(n, x):
%p A362995 seq(print(CoeffList(poly(n, x))), n = 0..7);
%o A362995 (SageMath)
%o A362995 def A362995row(n: int) -> list[int]:
%o A362995     s = add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
%o A362995         for j in (0..u)) for u in (0..n))
%o A362995     l = lcm(i + 1 for i in (0..n))
%o A362995     return (s * l).list()
%o A362995 for n in (0..7): print(A362995row(n))
%Y A362995 Cf. A362993 (row sums), A362994 (alternating row sums), A001008 (column 0), A362992 (main diagonal), A362996/A362997.
%Y A362995 Cf. A363000.
%K A362995 nonn,tabl
%O A362995 0,2
%A A362995 _Peter Luschny_, May 14 2023