A262014 Triangle in which the g.f. for row n is (1-x)^(4*n+1) * Sum_{j>=0} binomial(n+j-1,j)^4 * x^j, read by rows of k=0..3*n terms.

1, 1, 11, 11, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 243, 6750, 49682, 128124, 128124, 49682, 6750, 243, 1, 1, 608, 40136, 724320, 4961755, 15018688, 21571984, 15018688, 4961755, 724320, 40136, 608, 1, 1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275, 6021225, 167475, 1275, 1, 1, 2376, 554931, 35138736, 879018750, 10490842656, 66555527346, 239677178256, 509723668476, 654019630000, 509723668476

Offset: 0

Author

Paul D. Hanna, Sep 10 2015

Keywords

• nonn
• tabf

Examples

			Triangle begins:
 1;
 1, 11, 11, 1;
 1, 72, 603, 1168, 603, 72, 1;
 1, 243, 6750, 49682, 128124, 128124, 49682, 6750, 243, 1;
 1, 608, 40136, 724320, 4961755, 15018688, 21571984, 15018688, 4961755, 724320, 40136, 608, 1;
 1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275, 6021225, 167475, 1275, 1;
 1, 2376, 554931, 35138736, 879018750, 10490842656, 66555527346, 239677178256, 509723668476, 654019630000, 509723668476, 239677178256, 66555527346, 10490842656, 879018750, 35138736, 554931, 2376, 1;
 ...
Row g.f.s begin:
 n=0: (1) = (1-x) * (1 + x + x^2 + x^3 + x^4 +...);
 n=1: (1 + 11*x + 11*x^2 + x^3)  =  (1-x)^5 * (1 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 +...);
 n=2: (1 + 72*x + 603*x^2 + 1168*x^3 + 603*x^4 + 72*x^5 + x^6)  =  (1-x)^9 * (1 + 3^4*x + 6^4*x^2 + 10^4*x^3 + 15^4*x^5 + 21^4*x^6 +...);
 n=3: (1 + 243*x + 6750*x^2 + 49682*x^3 + 128124*x^4 + 128124*x^5 + 49682*x^6 + 6750*x^7 + 243*x^8 + x^9)  =  (1-x)^13 * (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 + 84^4*x^6 +...);
 ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1425, as a flattened triangle of rows 0..30
• Ilia Gaiur, Vladimir Rubtsov, and Duco van Straten, Product formulas for the Higher Bessel functions, arXiv:2405.03015 [math.AG], 2024. See p. 18.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.

Crossrefs

Cf. A008977 (row sums), A262015 (diagonal), A202750, A258402.

Cf. A181544 (triangle variant).

Programs

PARI

{T(n, k)=polcoeff(sum(j=0, n+k, binomial(n+j, j)^4*x^j)*(1-x)^(4*n+1), k)}
for(n=0, 10, for(k=0, 3*n, print1(T(n, k), ", ")); print(""))

Formula

Row sums form A008977(n) = (4*n)!/(n!)^4.

T(n,1) = A258402(n) = (n^2 + 4*n + 6) * n^2.

From Sergii Voloshyn, Dec 17 2024: (Start)

Let E be the operator D*x*D*x*D*x*D, where D denotes the derivative operator d/dx. Then (1/(n)!^4) * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(4*n+1) = Sum_{j>=0} binomial(n+j,j)^4 * x^j.

For example, when n = 2 we have (1/2!)^4*E^3(1/(1 - x)) = (1 + 243 x + 6750 x^2 + 49682 x^3 + 128124 x^4 + 128124 x^5 + 49682 x^6 + 6750 x^7 + 243 x^8 + x^9)/(1 - x)^13. (End)