A322892 a(n) = [x^n] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.

1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, 44097148953, 1092097482333, 44645622936189, 1338624157833861, 62791851488870493, 2213430779241737793, 117082536584478235713, 4748345510312622896993, 279463602946698380026793, 12824987274099379222626701, 830920299335152521399853101, 42586722790649923167650932101, 3011022417317079016258969826109, 170527854080899363788154404878305

Offset: 0

Author

Paul D. Hanna, Dec 29 2018

Keywords

• nonn

Examples

			The irregular triangle A322891 of coefficients of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
1;
1, 1, 2;
2, 3, 9, 6, 8;
6, 11, 42, 45, 84, 44, 48;
24, 50, 227, 310, 717, 620, 908, 400, 384;
120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
in which the main diagonal forms this sequence.
Note that the terms in the secondary diagonal A322893 in the above triangle
[1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...]
may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
[1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
		

Links

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

Crossrefs

Cf. A322891, A322893, A322894.

Cf. A322238 (variant).

Programs

PARI

{A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
/* Print the irregular triangle */
for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
/* Print this sequence */
for(n=0, 30, print1( A322891(n, n), ", "))

Formula

a(n+1) = 4*(n+1) * A322893(n) + a(n), for n >= 1.

a(n+1) = 2*n*(n+1)^2 * A322894(n) + a(n), for n >= 1.