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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.
4
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.
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
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Sep 03 2017
STATUS
approved