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

1,6

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

Links

Table of n, a(n) for n=1..81.

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

Example

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,
...

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 *)

Prog

(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

Crossrefs

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

Cf. A088459 (even numbered rows of A088855).

Sequence in context: A132823 A059317 A322046 * A220886 A256156 A342060

Adjacent sequences: A247641 A247642 A247643 * A247645 A247646 A247647

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 23 2014

Extensions

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

Offset changed to 1 by Georg Fischer, Sep 27 2021

Status

approved