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)).
Original entry on oeis.org
1, 3, 2, 11, 28, 18, 25, 184, 351, 192, 137, 2608, 11097, 16128, 7500, 147, 6816, 57591, 166912, 193750, 77760, 1089, 118464, 1865511, 9588736, 20843750, 20062080, 7058940, 2283, 567936, 16015401, 136921088, 495546875, 858003840, 704129265, 220200960
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 3, 2;
[2] 11, 28, 18;
[3] 25, 184, 351, 192;
[4] 137, 2608, 11097, 16128, 7500;
[5] 147, 6816, 57591, 166912, 193750, 77760;
[6] 1089, 118464, 1865511, 9588736, 20843750, 20062080, 7058940;
[7] 2283, 567936, 16015401, 136921088, 495546875, 858003840, 704129265, 220200960;
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R := (n, x) -> add(add(x^j*binomial(u, j)*(j + 1)^n, j = 0..u)/(u + 1), u=0..n):
CoeffList := p -> PolynomialTools:-CoefficientList(p, x):
poly := (n, x) -> ilcm(seq(i, i = 1..n+1)) * R(n, x):
seq(print(CoeffList(poly(n, x))), n = 0..7);
-
def A362995row(n: int) -> list[int]:
s = add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
for j in (0..u)) for u in (0..n))
l = lcm(i + 1 for i in (0..n))
return (s * l).list()
for n in (0..7): print(A362995row(n))
A363154
Triangle read by rows. The Hadamard product of A173018 and A349203.
Original entry on oeis.org
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
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;
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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;
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).
Original entry on oeis.org
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
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;
- 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.
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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;
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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 *)
-
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)
Showing 1-3 of 3 results.
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