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

0,6

Links

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

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

Example

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; ...

Prog

(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(""))

Crossrefs

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

Cf. A201949 (variant), A324956.

Sequence in context: A096646 A306234 A290057 * A302713 A136206 A262979

Adjacent sequences: A249787 A249788 A249789 * A249791 A249792 A249793

Keyword

nonn,tabf

Author

Paul D. Hanna, Nov 05 2014

Status

approved