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

Offset: 0

Author

Peter Luschny, May 16 2023

Keywords

• sign
• tabl

Comments

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

Examples

			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;
		

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.

Crossrefs

Variant: A051714/A051715.

Cf. A362994 (column 0), A002944 (main diagonal), A164555/A027642 (Bernoulli).

Programs

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 *)

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)

Formula

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