A322235 Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + k*x^2), for n >= 0.

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, 362880, 1026576, 4438620, 8936288, 21541905, 33149481, 56464695, 68597418, 89489025, 86715299, 89489025, 68597418, 56464695, 33149481, 21541905, 8936288, 4438620, 1026576, 362880

Offset

0,5

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5040, as a flattened triangle of rows 0..70.

Formula

Row sums equal (2*n+1)!/(n!*2^n), the odd double factorials.

Left and right borders equal n!.

Example

This irregular triangle 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 in the above triangle
[1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
may be divided by triangular numbers to obtain A322236:
[1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].

Mathematica

row[n_] := CoefficientList[Product[k+x+k*x^2, {k, 1, n}] + O[x]^(2n+1), x];
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Dec 26 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(""))

Crossrefs

Cf. A322236, A322237, A322238.

Cf. A322225 (variant), A322891 (variant).

Sequence in context: A321781 A254862 A340641 * A346477 A337583 A172984

Adjacent sequences: A322232 A322233 A322234 * A322236 A322237 A322238

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 15 2018

Status

approved