A291845 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + (2*k+1)*x + x^2) for n>0 with a single '1' in row 0.

1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 9, 26, 33, 26, 9, 1, 1, 16, 90, 224, 283, 224, 90, 16, 1, 1, 25, 235, 1050, 2389, 2995, 2389, 1050, 235, 25, 1, 1, 36, 511, 3660, 14174, 30324, 37723, 30324, 14174, 3660, 511, 36, 1, 1, 49, 980, 10339, 62265, 218246, 446109, 551047, 446109, 218246, 62265, 10339, 980, 49, 1, 1, 64, 1716, 25088, 218330, 1162560, 3782064, 7460928, 9157923, 7460928, 3782064, 1162560, 218330, 25088, 1716, 64, 1, 1, 81, 2805, 54324, 646542, 4899258, 23763914, 72918576, 139775763, 170606547, 139775763, 72918576, 23763914, 4899258, 646542, 54324, 2805, 81, 1

Offset: 0

Author

Paul D. Hanna, Sep 03 2017

Keywords

• nonn
• tabf

Comments

Row sums yield the odd double factorials A001147.

Central terms in rows form A291846.

Another diagonal forms A291847.

Antidiagonal sums yield A291848.

Examples

			This irregular triangle begins:
1;
1, 1, 1;
1, 4, 5, 4, 1;
1, 9, 26, 33, 26, 9, 1;
1, 16, 90, 224, 283, 224, 90, 16, 1;
1, 25, 235, 1050, 2389, 2995, 2389, 1050, 235, 25, 1;
1, 36, 511, 3660, 14174, 30324, 37723, 30324, 14174, 3660, 511, 36, 1;
1, 49, 980, 10339, 62265, 218246, 446109, 551047, 446109, 218246, 62265, 10339, 980, 49, 1;
1, 64, 1716, 25088, 218330, 1162560, 3782064, 7460928, 9157923, 7460928, 3782064, 1162560, 218330, 25088, 1716, 64, 1;
1, 81, 2805, 54324, 646542, 4899258, 23763914, 72918576, 139775763, 170606547, 139775763, 72918576, 23763914, 4899258, 646542, 54324, 2805, 81, 1;
1, 100, 4345, 107700, 1681503, 17237880, 117496358, 529332200, 1548992621, 2899264620, 3521075919, 2899264620, 1548992621, 529332200, 117496358, 17237880, 1681503, 107700, 4345, 100, 1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1680 of rows 0..40 of this triangle in flattened form.

Crossrefs

Cf. A291846, A291847, A291848, A201949, A001147 (row sums).

Programs

PARI

{T(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}
{for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))}