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
Keywords
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.
Programs
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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
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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
Comments