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

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn

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

Examples

			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,...)
		

Crossrefs

Cf. A192232, A192457.

Programs

Mathematica

(See A192457.)

Formula

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