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

Author

Robin Hankin, Oct 16 2019

Keywords

• nonn

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.

Crossrefs

Cf. A035610, A201805.

Programs

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

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

R

library("freealg")
g <- function(p,string){constant(as.freealg(string)^p)} sapply(0:7,g,"1+x+y+X+Y")

Formula

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

Extensions

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