A349203 -id:A349203 - cp's OEIS frontend

A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

1, 1, 0, 2, 1, 0, 3, 4, 1, 0, 12, 33, 22, 3, 0, 10, 52, 66, 26, 2, 0, 60, 570, 1208, 906, 228, 10, 0, 105, 1800, 5955, 7248, 3573, 600, 15, 0, 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0, 252, 14056, 102256, 264702, 312380, 176468, 43824, 3514, 28, 0

Offset: 0

Peter Luschny, May 21 2023

			Triangle T(n, k) starts:
[0]   1;
[1]   1,    0;
[2]   2,    1,     0;
[3]   3,    4,     1,     0;
[4]  12,   33,    22,     3,     0;
[5]  10,   52,    66,    26,     2,     0;
[6]  60,  570,  1208,   906,   228,    10,    0;
[7] 105, 1800,  5955,  7248,  3573,   600,   15,  0;
[8] 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0;

Peter Luschny, A combinatorial triangle for the Bernoulli numbers, MathOverflow 2023.

Cf. A173018, A349203, A002944 (column 0), A099946, A362994 (alternating row sums), A362990 (row sums).

A173018 := (n, k) -> combinat[eulerian1](n, k):
A349203 := (n, k) -> ilcm(seq(binomial(n, j), j = 0..n)) / binomial(n, k):
A363154 := (n, k) -> A173018(n, k) * A349203(n, k):
for n from 0 to 8 do seq(A363154(n, k), k = 0..n) od;

T(n, k) = A173018(n, k) * A349203(n, k).

Sum_{k=0..n} (-1)^k * T(n, k) = lcm(1, 2, ..., n+1)*Bernoulli(n, 1) = A362994(n).

A362990 Row sums of A363154.

Original entry on oeis.org

1, 1, 3, 8, 70, 156, 2982, 19296, 216396, 917480, 47847580, 250815120, 18724750980, 116905717320, 788402337660, 11422960203648, 1503958432951464, 12400297557081840, 2060467159019459400, 19050402239258292000, 185663459342909159640, 1902329957427442126320

Offset: 0

Peter Luschny, May 22 2023

nonn

Cf. A363154, A173018, A349203.

A362990 := n -> add(ilcm(seq(binomial(n, j), j = 0..n)) * combinat[eulerian1](n, k) / binomial(n, k), k = 0.. n): seq(A362990(n), n = 0..21);

cp's OEIS Frontend

A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

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