A054649 -id:A054649 - cp's OEIS frontend

A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

1, 1, 1, 1, 1, 2, 1, 0, 5, 6, 1, -2, 11, 14, 24, 1, -5, 25, 5, 94, 120, 1, -9, 55, -75, 304, 444, 720, 1, -14, 112, -350, 1099, 364, 3828, 5040, 1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320, 1, -27, 366, -2646, 12873, -31563, 79064, 34236, 270576, 362880

Offset: 0

N. J. A. Sloane, Apr 17 2000

Apparently A190782 with reversed rows. - Mathew Englander, May 17 2014

			The first few polynomials are:
  1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
So the triangle begins:
  1;
  1,   1;
  1,   1,   2;
  1,   0,   5,     6;
  1,  -2,  11,    14,   24;
  1,  -5,  25,     5,   94,   120;
  1,  -9,  55,   -75,  304,   444,   720;
  1, -14, 112,  -350, 1099,   364,  3828,  5040;
  1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

T(2*n,n) gives A347987.

Cf. A054649, A054655, A054654, A190782 .

c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 9}] // Flatten  (* Jean-François Alcover, Oct 04 2012 *)

T(n, k) = Sum_{i=0..k} Stirling1(i+n-k,n-k)*n!/(i+n-k)!. - Igor Victorovich Statsenko, May 27 2024

Missing 0 corrected by Steve Marak - N. J. A. Sloane, Jul 27 2012

A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769

Offset: 0

N. J. A. Sloane, Apr 18 2000

			Triangle begins:
  1;
  1,  -1;
  1,  -5,    6;
  1, -12,   47,    -60;
  1, -22,  179,   -638,    840;
  1, -35,  485,  -3325,  11274,   -15120;
  1, -51, 1075, -11985,  74524,  -245004,  332640;
  1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened).
Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

Cf. A000407, A054649, A054651, A054654, A008276.

a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010

row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Apr 06 2012, after Maple *)

T(n,k)=polcoef(n!*binomial(x-n,n), n-k);

n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
From Robert Israel, Jul 12 2016: (Start)
G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k  = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).
E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)

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

Original entry on oeis.org

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

A347985 a(n) = [x^n] (2n)! Sum_{k=0..2n} binomial(x-2n,k).

Original entry on oeis.org

1, -3, 131, -8955, 893249, -117408375, 19180128407, -3747886705563, 852713408774513, -221431482383149467, 64629367172619230475, -20945446993569455512155, 7463226827397324296491489, -2899926767958744905966692575

Offset: 0

Seiichi Manyama, Sep 24 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..283

Cf. A054649.

a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x-2*n, k)), n);

a(n) = A054649(2*n,n).

cp's OEIS Frontend

A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

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A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

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

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A347985 a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x-2*n,k).

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Author

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Programs

PARI

Formula

A347985 a(n) = [x^n] (2n)! Sum_{k=0..2n} binomial(x-2n,k).