A291846 -id:A291846 - cp's OEIS frontend

A291845 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + (2k+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

Paul D. Hanna, Sep 03 2017

Row sums yield the odd double factorials A001147.
Central terms in rows form A291846.
Another diagonal forms A291847.
Antidiagonal sums yield A291848.

			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; ...

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

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

{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(""))}

A291847 A diagonal of irregular triangle A291845.

Original entry on oeis.org

1, 4, 26, 224, 2389, 30324, 446109, 7460928, 139775763, 2899264620, 65954625560, 1632654953280, 43688087178059, 1256602120453484, 38661480001233486, 1266934683224418816, 44054989554206606603, 1620147926716343851500, 62826169539072988352134, 2562071016044926371845920, 109611597248567432265872903, 4908887251696851858305862332

Offset: 0

Paul D. Hanna, Sep 03 2017

nonn

G.f. of row n in triangle A291845 equals Product_{k=0..n-1} (1 + (2*k+1)*x + x^2), with row sums equal to the odd double factorials A001147.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A291845, A291846, A291848.

/* As a Diagonal in Triangle A291845 */
{A291845(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}
{for(n=0,25,print1(A291845(n+1,n),", "))}

a(n) = A291845(n+1,n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A291847 A diagonal of irregular triangle A291845.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A291847 A diagonal of irregular triangle A291845.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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