A098118 -id:A098118 - cp's OEIS frontend

A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).

1, 2, 1, 12, 7, 1, 120, 74, 15, 1, 1680, 1066, 251, 26, 1, 30240, 19524, 5000, 635, 40, 1, 665280, 434568, 117454, 16815, 1345, 57, 1, 17297280, 11393808, 3197348, 495544, 45815, 2527, 77, 1, 518918400, 343976400, 99236556, 16275700, 1659889, 107800, 4354, 100, 1

Offset: 0

Peter Bala, Nov 05 2015

The row polynomials are a Sheffer sequence. For the associated polynomial sequence of binomial type see A038455.

			Triangle begins:
[0]       1;
[1]       2,      1;
[2]      12,      7,      1;
[3]     120,     74,     15,     1;
[4]    1680,   1066,    251,    26,    1;
[5]   30240,  19524,   5000,   635,   40,  1;
[6]  665280, 434568, 117454, 16815, 1345, 57, 1;
  ...

R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, Section 5.6 CreateSpace Independent Publishing Platform 2006, ISBN-13: 978-1502925244.
Wikipedia, Sheffer sequence

Cf. A001813 (column 0), A005449 (first subdiagonal), A098118 (column 1).
Cf. A006963 (row sums), A000407 (alternating row sum).
Cf. A054649, A000108, A000984, A038455, A092932.

seq(seq(coeff(product(n + x + k, k = 1 .. n), x, i), i = 0..n), n = 0..8);
# Alternative:
p := n -> n!*hypergeom([-n, -x + n], [-n], 1):
seq(seq((-1)^k*coeff(simplify(p(n)), x, k), k=0..n), n=0..6); # Peter Luschny, Nov 27 2021

p[n_, x_] := FunctionExpand[Gamma[2*n + x + 1] / Gamma[n + x + 1]];
Table[CoefficientList[p[n, x], x], {n,0,8}] // Flatten (* Peter Luschny, Mar 21 2022 *)

E.g.f.: A(x,t) = B(t)*C(t)^x = 1 + (2 + x)*t + (3 + x)*(4 + x)*t^2/2! + (4 + x)*(5 + x)*(6 + x)*t^3/3! + ..., where B(t) = 1/sqrt(1 - 4*t) is the o.g.f. for A000984 and C(t) = (1 - sqrt(1 - 4*t))/(2*t) is the o.g.f. for A000108.
n-th row polynomial: n!*binomial(2*n + x,n).
T(n, k) = (-1)^k*n!*[x^k] hypergeom([-n, -x + n], [-n], 1). - Peter Luschny, Nov 27 2021
T(n, k) = [x^k] Gamma(2*n + x + 1) / Gamma(n + x + 1). - Peter Luschny, Mar 21 2022

A384136 a(n) = (3n)!/(2n)! * Sum_{k=1..n} 1/(2*n+k).

Original entry on oeis.org

1, 11, 191, 4578, 140274, 5238132, 230784840, 11720201616, 674092013040, 43310839531680, 3074579815271040, 238983481496188800, 20187063842072319360, 1841332369689189619200, 180372122189263722009600, 18885338733119777188300800, 2104722524872544008142592000

Offset: 1

Seiichi Manyama, May 20 2025

nonn

Cf. A098118, A384137.

Cf. A383678.

a(n) = sum(k=0, n, k*(2*n+1)^(k-1)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k * (2*n+1)^(k-1) * |Stirling1(n,k)|.

a(n) = n! * [x^n] (-log(1 - x)/(1 - x)^(2*n+1)).

A384167 a(n) = 2^n * n! * binomial(3n/2,n) Sum_{k=1..n} 1/(n+2*k).

Original entry on oeis.org

1, 10, 143, 2736, 66009, 1926912, 66086271, 2605455360, 116123049585, 5774107852800, 316921177332495, 19032668386099200, 1241454631056114825, 87402945316493721600, 6606130538582006306175, 533534147838972474163200, 45855293972076668267481825, 4178822478568980876361728000

Offset: 1

Seiichi Manyama, May 21 2025

nonn

Cf. A098118, A384168, A384169.

Cf. A113551.

a(n) = sum(k=0, n, k*(n+2)^(k-1)*2^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k * (n+2)^(k-1) * 2^(n-k) * |Stirling1(n,k)|.

a(n) = n! * [x^n] ( -log(1 - 2*x)/(2 * (1 - 2*x)^(n/2+1)) ).

A384168 a(n) = 3^n * n! * binomial(4n/3,n) Sum_{k=1..n} 1/(n+3*k).

Original entry on oeis.org

1, 13, 234, 5566, 165944, 5966136, 251491120, 12169996912, 665146831680, 40530954643840, 2724842629685120, 200361647815660800, 15997170878205905920, 1378271357428552115200, 127459020533529062246400, 12593128815600367187507200, 1323895109721239722075136000

Offset: 1

Seiichi Manyama, May 21 2025

nonn

Cf. A098118, A384167, A384169.

Cf. A303486.

a(n) = sum(k=0, n, k*(n+3)^(k-1)*3^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k * (n+3)^(k-1) * 3^(n-k) * |Stirling1(n,k)|.

a(n) = n! * [x^n] ( -log(1 - 3*x)/(3 * (1 - 3*x)^(n/3+1)) ).

A384169 a(n) = 4^n * n! * binomial(5n/4,n) Sum_{k=1..n} 1/(n+4*k).

Original entry on oeis.org

1, 16, 347, 9856, 349269, 14885760, 742589175, 42479124480, 2742327328905, 197267905658880, 15649214440432275, 1357388618032742400, 127808331929417605725, 12983375200126773657600, 1415428114244995252270575, 164837363498660501913600000, 20423530465926352502482292625

Offset: 1

Seiichi Manyama, May 21 2025

nonn

Cf. A098118, A384167, A384168.

Cf. A303487.

a(n) = sum(k=0, n, k*(n+4)^(k-1)*4^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k * (n+4)^(k-1) * 4^(n-k) * |Stirling1(n,k)|.

a(n) = n! * [x^n] ( -log(1 - 4*x)/(4 * (1 - 4*x)^(n/4+1)) ).

A253669 Square array read by ascending antidiagonals, T(n, k) = k![x^k](log(x+1)sum(j=0..n, C(2n,j)x^j)), n>=0, k>=0.

Original entry on oeis.org

0, 0, 1, 0, 1, -1, 0, 1, 3, 2, 0, 1, 7, -4, -6, 0, 1, 11, 26, 10, 24, 0, 1, 15, 74, -46, -36, -120, 0, 1, 19, 146, 342, 144, 168, 720, 0, 1, 23, 242, 1066, -756, -624, -960, -5040, 0, 1, 27, 362, 2414, 5944, 2844, 3408, 6480, 40320, 0, 1, 31, 506, 4578, 19524

Offset: 0

Peter Luschny, Jan 18 2015

			Square array starts:
[n\k][0   1   2    3     4      5       6]
[0]   0,  1, -1,   2,   -6,    24,   -120, ...
[1]   0,  1,  3,  -4,   10,   -36,    168, ...
[2]   0,  1,  7,  26,  -46,   144,   -624, ...
[3]   0,  1, 11,  74,  342,  -756,   2844, ...
[4]   0,  1, 15, 146, 1066,  5944, -15768, ...
[5]   0,  1, 19, 242, 2414, 19524, 127860, ...
[6]   0,  1, 23, 362, 4578, 48504, 434568, ...
The first few rows as a triangle:
0,
0, 1,
0, 1, -1,
0, 1,  3,   2,
0, 1,  7,  -4,  -6,
0, 1, 11,  26,  10,  24,
0, 1, 15,  74, -46, -36, -120,
0, 1, 19, 146, 342, 144,  168, 720.

Cf. A098118.

T := (n,k) -> k!*coeff(series(ln(x+1)*add(binomial(2*n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k), k=0..6)) od;

T(n,n) = A098118(n).

cp's OEIS Frontend

A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).

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Maple

Mathematica

Formula

A384136 a(n) = (3*n)!/(2*n)! * Sum_{k=1..n} 1/(2*n+k).

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PARI

Formula

A384167 a(n) = 2^n * n! * binomial(3*n/2,n) * Sum_{k=1..n} 1/(n+2*k).

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PARI

Formula

A384168 a(n) = 3^n * n! * binomial(4*n/3,n) * Sum_{k=1..n} 1/(n+3*k).

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PARI

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A384169 a(n) = 4^n * n! * binomial(5*n/4,n) * Sum_{k=1..n} 1/(n+4*k).

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PARI

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A253669 Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A384136 a(n) = (3n)!/(2n)! * Sum_{k=1..n} 1/(2*n+k).

A384167 a(n) = 2^n * n! * binomial(3n/2,n) Sum_{k=1..n} 1/(n+2*k).

A384168 a(n) = 3^n * n! * binomial(4n/3,n) Sum_{k=1..n} 1/(n+3*k).

A384169 a(n) = 4^n * n! * binomial(5n/4,n) Sum_{k=1..n} 1/(n+4*k).

A253669 Square array read by ascending antidiagonals, T(n, k) = k![x^k](log(x+1)sum(j=0..n, C(2n,j)x^j)), n>=0, k>=0.