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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).
2
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
0,5
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.
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)
CROSSREFS
Variant: A051714/A051715.
Cf. A362994 (column 0), A002944 (main diagonal), A164555/A027642 (Bernoulli).
Sequence in context: A370885 A072738 A165316 * A215976 A141058 A102706
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, May 16 2023
STATUS
approved