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

1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, 39093550782, 872184375426, 24255771626177, 637337897821275, 20280636608372520, 613459886387884248, 21980602229339160099, 752319889132243511097, 29971937072992355586420, 1145176991891992950865380, 50212935598435298798304888, 2118631687709497814282735724, 101386084189432812756067599036

Offset

1,2

Comments

a(n) = n*(n+1)/2 * A322236(n) for n >= 1.

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 this sequence forms a diagonal.

Mathematica

a[n_] := SeriesCoefficient[Product[k + x + k x^2, {k, 1, n}], {x, 0, n-1}];
Array[a, 23] (* 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), ", "))

Crossrefs

Cf. A322235, A322238, A322236.

Sequence in context: A240916 A006292 A067370 * A289795 A094432 A354677

Adjacent sequences: A322234 A322235 A322236 * A322238 A322239 A322240

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2018

Status

approved