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

Author

Peter Bala, Nov 05 2015

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A001813 (column 0), A005449 (first subdiagonal), A098118 (column 1).

Cf. A006963 (row sums), A000407 (alternating row sum).

Cf. A054649, A000108, A000984, A038455, A092932.

Programs

Maple

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

Mathematica

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 *)

Formula

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