A322236 a(n) = A322237(n) / (n*(n+1)/2), where A322237(n) = [x^(n-1)] Product_{k=1..n} (k + x + k*x^2), for n >= 1.

1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, 11181850967, 266546940947, 6069884741155, 169005305069771, 4510734458734443, 143664066858425883, 4399531515393236907, 157747037226275555718, 5453223770914252146978, 217372015577641986139848, 8374038291341888594002908, 367340884744321785348071011, 15606634300050239405862650475

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..301

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 A322238.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal (A322237), beginning
[1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
may be divided by triangular numbers to obtain this sequence
[1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].

Mathematica

a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}]/ (n(n+1)/2);
Array[a, 24] (* Jean-François Alcover, Dec 28 2018 *)

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=1, 30, print1( T(n, n-1)/(n*(n+1)/2), ", "))

Crossrefs

Cf. A322237, A322238.

Sequence in context: A306561 A226102 A094356 * A318152 A358083 A323552

Adjacent sequences: A322233 A322234 A322235 * A322237 A322238 A322239

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2018

Status

approved