A322228 a(n) = [x^n] Product_{k=1..n} (k + x - k*x^2), for n >= 0.

1, 1, -3, -21, 75, 1475, -5005, -221389, 593523, 57764619, -89101881, -23273632371, 953636541, 13409519997705, 23908442020749, -10469975115603501, -40844292735050541, 10646036726696597027, 66995992524016223543, -13672657170891872702719, -122282221141986787179519, 21647316686778755963070321, 256325163531592225309743129, -41426918732532942751217361155, -620418821801458605268716606275, 94275566307675915918535250768725

Offset: 0

Author

Paul D. Hanna, Dec 15 2018

Keywords

• sign

Comments

a(n+1) = -2*(n+1) * A322227(n) + a(n), for n >= 1.

a(n+1) = -n*(n+1)^2 * A322226(n) + a(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 the central terms equal this sequence.
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 A322226:
[1, 1, -2, -14, 36, 736, -1819, -88227, 163185, 19160769, -15294993, ...].
		

Links

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

Crossrefs

Cf. A322235, A322236, A322237.

Cf. A322238 (variant).

Programs

Mathematica

a[n_] := SeriesCoefficient[Product[k + x - k x^2, {k, 1, n}], {x, 0, n}];
Array[a, 26, 0] (* 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=0, 30, print1( T(n, n), ", "))