A362991 - OEIS

A362991

Triangle read by rows. T(n, k) = lcm{1, 2, ..., n+1} * Sum_{j=0..n-k} (-1)^(n-k-j) * j! * Stirling2(n - k, j) / (j + k + 1).

1, 1, 1, 1, 2, 2, 0, 2, 3, 3, -2, 2, 9, 12, 12, 0, -2, 3, 8, 10, 10, 10, -10, -9, 24, 50, 60, 60, 0, 20, -30, -8, 50, 90, 105, 105, -84, 84, 18, -96, 0, 150, 245, 280, 280, 0, -84, 126, -24, -90, 18, 147, 224, 252, 252, 2100, -2100, 126, 1344, -600, -870, 343, 1568, 2268, 2520, 2520

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OFFSET

0,5

COMMENTS

A variant of the Akiyama-Tanigawa algorithm for the Bernoulli numbers A164555/ A027642.

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..11324 (rows 0..150 of the triangle, flattened)

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Index entries for sequences related to Bernoulli numbers.

FORMULA

T(n, 0) = lcm(1, 2, ..., n+1) * Bernoulli(n, 1).

EXAMPLE

Triangle T(n, k) starts:

[0] 1;

[1] 1, 1;

[2] 1, 2, 2;

[3] 0, 2, 3, 3;

[4] -2, 2, 9, 12, 12;

[5] 0, -2, 3, 8, 10, 10;

[6] 10, -10, -9, 24, 50, 60, 60;

[7] 0, 20, -30, -8, 50, 90, 105, 105;

[8] -84, 84, 18, -96, 0, 150, 245, 280, 280;

[9] 0, -84, 126, -24, -90, 18, 147, 224, 252, 252;

MAPLE

LCM := n -> ilcm(seq((1 + i), i = 0..n)):

T := (n, k) -> LCM(n)*add((-1)^(n - k - j)*j!*Stirling2(n - k, j)/(j + k + 1), j = 0..n - k):

for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

MATHEMATICA

A362991row[n_]:=Table[LCM@@Range[n+1]Sum[(-1)^(n-k-j)j!StirlingS2[n-k, j]/(j+k+1), {j, 0, n-k}], {k, 0, n}]; Array[A362991row, 15, 0] (* Paolo Xausa, Aug 09 2023 *)

PROG

(SageMath)

def A362991Triangle(size): # 'size' is the number of rows.

A, T, l = [], [], 1

for n in range(size):

A.append(Rational(1/(n + 1)))

for j in range(n, 0, -1):

A[j - 1] = j * (A[j - 1] - A[j])

l = lcm(l, n + 1)

T.append([a * l for a in A])

return T

A362991Triangle(10)