A249790 Triangle in which row n lists the coefficients in Product_{k=1..n} (1 + k*x + x^2), for n>=0, as read by rows.

1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 6, 14, 18, 14, 6, 1, 1, 10, 39, 80, 100, 80, 39, 10, 1, 1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1, 1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1, 1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1, 1, 36, 554, 4788, 25753, 90720, 216166, 358056, 422252

Offset: 0

Author

Paul D. Hanna, Nov 05 2014

Keywords

• nonn
• tabf

Examples

			Triangle begins:
1;
1, 1, 1;
1, 3, 4, 3, 1;
1, 6, 14, 18, 14, 6, 1;
1, 10, 39, 80, 100, 80, 39, 10, 1;
1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1;
1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1;
1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1;
1, 36, 554, 4788, 25753, 90720, 216166, 358056, 422252, 358056, 216166, 90720, 25753, 4788, 554, 36, 1;
1, 45, 879, 9810, 69399, 327285, 1058399, 2394270, 3860922, 4516380, 3860922, 2394270, 1058399, 327285, 69399, 9810, 879, 45, 1;
1, 55, 1330, 18645, 168378, 1031085, 4400648, 13305545, 28862021, 45519870, 52885644, 45519870, 28862021, 13305545, 4400648, 1031085, 168378, 18645, 1330, 55, 1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1088, listing terms in rows 0..32 of flattened triangle.

Crossrefs

Cf. A201826 (central coefficients), A202474 (a diagonal), A202476, A001710 (row sums).

Cf. A201949 (variant), A324956.

Programs

PARI

{T(n,k)=polcoeff(prod(m=1, n, 1 + m*x + x^2 +x*O(x^k)), k,x)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

Formula

E.g.f.: 1/(1 - x*y)^(1/y + 1 + y). - Paul D. Hanna, Mar 02 2019

E.g.f.: A(x,y) = 1/(1-x*y) * Sum_{k>=0} (1/y^k + y^k)/2^(0^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!). - Paul D. Hanna, Mar 02 2019

E.g.f. of diagonal k: (1/y^k)/(1-x*y) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - Paul D. Hanna, Mar 02 2019

E.g.f.: A(x,y) = x / Series_Reversion( F(x,y) ) such that F(x/A(x,y),y) = x, where F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (n+k)*y + n*y^2). - Paul D. Hanna, Mar 02 2019