A328494 Constant term in the expansion of (1+x+y+1/x+1/y)^n without assuming commutativity.

1, 1, 5, 13, 53, 181, 713, 2689, 10661, 41989, 168785, 680329, 2770409, 11331529, 46639157, 192762013, 800228069, 3333843685, 13936599857, 58432259977, 245665962113, 1035412181761, 4373982501245, 18516210906853, 78536526586553, 333712398776281, 1420364536094093

Offset

0,3

Comments

Related to A035610 which is the constant term of (x+y+1/x+1/y)^(2n).

If commutativity is assumed then the constant term of (1+x+y+1/x+1/y)^n is given by A201805(n). - Andrew Howroyd, Oct 25 2019

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Mark Haiman, Non-commutative rational power series and algebraic generating functions, European Journal of Combinatorics, 14(4):335-9 (1993).

Robin Hankin, Discussion of this and similar sequences

Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A035610(k). - Andrew Howroyd, Oct 25 2019

Maple

h := n -> GAMMA(n+1/2)/GAMMA(n+2)*hypergeom([2, 1-n], [n+2], -3):
a := n -> 3-(-3)^n-5^n+(1/sqrt(Pi))*add(12^(k+1)*binomial(n, 2*k)*h(k), k=1..n/2):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 25 2019

Prog

(R)
library("freealg")
g <- function(p, string){constant(as.freealg(string)^p)} sapply(0:7, g, "1+x+y+X+Y")
(PARI) a(n)={my(p=3/(1+2*sqrt(1-12*x+O(x*x^(n\2))))); sum(k=0, n\2, binomial(n, 2*k)*polcoef(p, k))} \\ Andrew Howroyd, Oct 25 2019

Crossrefs

Cf. A035610, A201805.

Sequence in context: A007231 A158345 A372370 * A149540 A149541 A149542

Adjacent sequences: A328491 A328492 A328493 * A328495 A328496 A328497

Keyword

nonn

Author

Robin Hankin, Oct 16 2019

Extensions

Terms a(8) and beyond from Andrew Howroyd, Oct 25 2019

Status

approved