A247644 Triangle formed from the odd-numbered rows of A088855.

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1, 1, 7, 49, 147, 441, 735, 1225, 1225, 1225, 735, 441, 147, 49, 7, 1, 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900, 3920, 3136, 1568, 784, 224, 64, 8, 1

Offset: 1

Author

N. J. A. Sloane, Sep 23 2014

Keywords

• nonn
• tabf

Comments

The rows give the coefficients in the numerator polynomials of the o.g.f.s for the columns of triangle A055898. - Georg Fischer, Aug 16 2021

They also occur (with a factor 2*x) in the numerator polynomials of the difference A157052-A157074. - Georg Fischer, Sep 27 2021

Examples

			Triangle begins:
1,
1,1,1,
1,2,4,2,1,
1,3,9,9,9,3,1,
1,4,16,24,36,24,16,4,1,
1,5,25,50,100,100,100,50,25,5,1,
1,6,36,90,225,300,400,300,225,90,36,6,1,
1,7,49,147,441,735,1225,1225,1225,735,441,147,49,7,1,
1,8,64,224,784,1568,3136,3920,4900,3920,3136,1568,784,224,64,8,1,
...
		

Links

• Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Crossrefs

Cf. A055899, A055900, A055901, A055902, A055903, A055904, A088855.

Cf. A088459 (even numbered rows of A088855).

Programs

Mathematica

row[n_] := CoefficientList[Sum[Binomial[n, k]^2 *x^(2*k), {k, 0, n}] + Sum[Binomial[n, k]*Binomial[n, k - 1]* x^(2*k - 1), {k, 0, n}], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 07 2018 *)

PARI

T(n, k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ A088855
row(n) = vector(2*n-1, k, T(2*n-1, k)); \\ Michel Marcus, Sep 27 2021

Extensions

Row n=8 corrected by Jean-François Alcover, Jun 07 2018

Offset changed to 1 by Georg Fischer, Sep 27 2021