A363000 -id:A363000 - cp's OEIS frontend

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)).

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

Peter Luschny, May 14 2023

			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;

Peter Luschny, An integer triangle for representing the Bernoulli numbers, Mathematics Stack Exchange, May 2023.
Index entries for sequences related to Bernoulli numbers.

Cf. A362993 (row sums), A362994 (alternating row sums), A001008 (column 0), A362992 (main diagonal), A362996/A362997.

Cf. A363000.

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))

T(n, k) = lcm(1,2, ..., n+1) * A362996(n, k) / A362997(n, k).

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

A362999 a(n) = denominator(R(2n + 1, 2n + 1, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Original entry on oeis.org

2, 3, 15, 105, 315, 3465, 45045, 9009, 153153, 14549535, 14549535, 19684665, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 316824683175, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725

Offset: 0

Peter Luschny, May 12 2023

See A363000 for a discussion of the polynomials.

Cf. A363001 (all denominators), A363000.

Maple
```
# See A363000 for a program.
```

A363001 a(n) = denominator(R(n, n, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Original entry on oeis.org

1, 2, 2, 3, 2, 15, 2, 105, 2, 315, 2, 3465, 2, 45045, 2, 9009, 2, 153153, 2, 14549535, 2, 14549535, 2, 19684665, 2, 1673196525, 2, 5019589575, 2, 145568097675, 2, 4512611027925, 2, 4512611027925, 2, 4512611027925, 2, 166966608033225, 2, 316824683175, 2, 6845630929362225

Offset: 0

Peter Luschny, May 12 2023

See A363000 for a discussion of the polynomials.

Cf. A363000 (numerators), A362999 (odd-indexed denominators).

Maple
```
# See A363000 for a program.
```

A362998 a(n) = Sum_{k=0..2n} R(2n, k, 1) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Original entry on oeis.org

1, 14, 867, 191476, 92323925, 78056422494, 102352057350319, 192403045957195112, 490874140662802917417, 1632516441577827373044370, 6861835036233587237791174211, 35570322051950868085847089286364, 222935341340851671535556256425205757, 1661810870170209385499116531813558149446

Offset: 0

Peter Luschny, May 12 2023

nonn

See A363000 for a discussion of the polynomials.

Cf. A363000.

Maple
```
# See A363000 for a program.
```

cp's OEIS Frontend

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)).

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Formula

A362999 a(n) = denominator(R(2n + 1, 2n + 1, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

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Maple

A363001 a(n) = denominator(R(n, n, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

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Programs

Maple

A362998 a(n) = Sum_{k=0..2n} R(2n, k, 1) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

cp's OEIS Frontend

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)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

SageMath

Formula

A362999 a(n) = denominator(R(2*n + 1, 2*n + 1, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

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Comments

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Programs

Maple

A363001 a(n) = denominator(R(n, n, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

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Comments

Crossrefs

Programs

Maple

A362998 a(n) = Sum_{k=0..2*n} R(2*n, k, 1) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A362999 a(n) = denominator(R(2n + 1, 2n + 1, 1)) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

A362998 a(n) = Sum_{k=0..2n} R(2n, k, 1) where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).