A322226 a(n) = A322227(n) / (n*(n+1)/2), where A322227(n) = [x^(n-1)] Product_{k=1..n} (k + x - k*x^2), for n >= 1.

1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, -6611719113, -4120456053, 3331391605595, 7909978546731, -2311176691053557, -10230565685787877, 2114443392338548619, 14290732101459857168, -2468208492961535459212, -23089123066195736850322, 3581650102772343613724618, 43704098963536055443651815, -6326399008631824968253597825, -96742680150222152446019734205

Offset: 1

Author

Paul D. Hanna, Dec 15 2018

Keywords

• sign

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 the central terms equal A322228.
RELATED SEQUENCES.
Note that the terms in the secondary diagonal A322227 in the above triangle
[1, 3, -12, -140, 540, 15456, -50932, -3176172, 7343325, 1053842295, ...]
may be divided by triangular numbers to obtain this sequence
[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. A322227, A322228.

Cf. A322236 (variant), A322894 (variant).

Programs

Mathematica

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

PARI

{A322225(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( A322225(n, k), ", ")); print(""))
/* Print this sequence */
for(n=1, 30, print1( A322225(n, n-1)/(n*(n+1)/2), ", "))