A322227 a(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.

1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, -1009469538, -515714090814, -374961500823, 349796118587475, 949197425607720, -314320029983283752, -1565276549925545181, 361569820089891813849, 2715239099277372861920, -518323783521922446434520, -5333587428291215212424382, 906157476001402934272328354, 12062331313935951302447900940, -1897919702589547490476079347500, -31441371048822199544956413616625

Offset: 1

Author

Paul D. Hanna, Dec 15 2018

Keywords

• sign

Comments

a(n) = n*(n+1)/2 * A322226(n) for n >= 1.

Examples

			The irregular triangle A322225 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, -3, -3, 2;
6, 11, -12, -21, 12, 11, -6;
24, 50, -61, -140, 75, 140, -61, -50, 24;
120, 274, -375, -1011, 540, 1475, -540, -1011, 375, 274, -120;
720, 1764, -2696, -8085, 4479, 15456, -5005, -15456, 4479, 8085, -2696, -1764, 720;
5040, 13068, -22148, -71639, 42140, 169266, -50932, -221389, 50932, 169266, -42140, -71639, 22148, 13068, -5040; ...
in which this sequence forms a diagonal.
RELATED SEQUENCES.
Note that the terms in this sequence
[1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
may be divided by triangular numbers n*(n+1)/2 to obtain A322226:
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].
		

Links

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

Crossrefs

Cf. A322225, A322226.

Programs

Mathematica

a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n-1}];
Array[a, 25] (* Jean-François Alcover, Dec 29 2018 *)

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), ", "))