A192459 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

1, 3, 17, 133, 1315, 15675, 218505, 3485685, 62607195, 1250116875, 27468111825, 658579954725, 17109329512275, 478744992200475, 14354443912433625, 459128747151199125, 15604187119787140875, 561558837528374560875, 21332903166207470462625

Offset

0,2

Comments

The polynomial p(n,x) is defined by recursively by p(n,x)=(x+2n)*p(n-1,x) with p[0,x]=x. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

Let transform T take the sequence {b(n), n>=1} to the sequence {c(n)} defined by: c(n) = det(M_n), where M_n denotes the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j. Conjecture: a(n) = abs(c(n+1)), where c(n) denotes transform T of triangular numbers (A000217). - Lechoslaw Ratajczak, Jul 26 2021

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = (1/3)*(2^(n+1)*(n+1)! + (2n-1)!!). - Vaclav Potocek, Feb 04 2016

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=x -> x
p(1,x)=x(2+x) -> 2+3x
p(2,x)=x(2+x)(4+x) -> 14+17x
p(3,x)=x(2+x)(4+x)(6+x) -> 118+133x.
From these, read
A192457=(1,2,14,118,...) and A192459=(1,3,17,133,...)

Mathematica

(See A192457.)

Crossrefs

Cf. A192232, A192457.

Sequence in context: A163684 A363135 A093986 * A362748 A055214 A105630

Adjacent sequences: A192456 A192457 A192458 * A192460 A192461 A192462

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved