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

0,6

Comments

Row sums yield the odd double factorials A001147.

Central terms in rows form A291846.

Another diagonal forms A291847.

Antidiagonal sums yield A291848.

Links

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

Example

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; ...

Prog

(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(""))}

Crossrefs

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

Sequence in context: A255698 A290558 A071992 * A322193 A174984 A092141

Adjacent sequences: A291842 A291843 A291844 * A291846 A291847 A291848

Keyword

nonn,tabf

Author

Paul D. Hanna, Sep 03 2017

Status

approved