A322238 a(n) = [x^n] Product_{k=1..n} (k + x + k*x^2), for n >= 0.

1, 1, 5, 23, 215, 1815, 24495, 302619, 5320667, 86715299, 1876495799, 38014052089, 976259270857, 23653053031933, 702814658564889, 19822951593203139, 668803323061123779, 21526439460249188211, 812828119716458951775, 29400983906741712373461, 1228278466826435935830261, 49325712126290139872176221, 2258694878457443286997591293, 99715752513094342744003434597, 4966247793605869355035248188325

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n+1) = 2*(n+1) * A322237(n) + a(n), for n >= 1.

a(n+1) = n*(n+1)^2 * A322236(n) + a(n), for n >= 1.

Example

The irregular triangle A322235 formed from coefficients of x^k in Product_{m=1..n} (m + x + m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 1;
2, 3, 5, 3, 2;
6, 11, 24, 23, 24, 11, 6;
24, 50, 131, 160, 215, 160, 131, 50, 24;
120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
in which the central terms equal this sequence.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322237 in the above triangle
[1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
may be divided by triangular numbers to obtain A322236:
[1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].

Prog

(PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
/* Print this sequence */
for(n=0, 30, print1( T(n, n), ", "))

Crossrefs

cF. A322235, A322236, A322237.

Cf. A322228 (variant), A322894 (variant).

Sequence in context: A203223 A173041 A156187 * A295080 A323993 A156555

Adjacent sequences: A322235 A322236 A322237 * A322239 A322240 A322241

Keyword

nonn

Author

Paul D. Hanna, Dec 12 2018

Status

approved