cp's OEIS Frontend

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

%I A328494 #32 Dec 03 2021 17:07:55
%S A328494 1,1,5,13,53,181,713,2689,10661,41989,168785,680329,2770409,11331529,
%T A328494 46639157,192762013,800228069,3333843685,13936599857,58432259977,
%U A328494 245665962113,1035412181761,4373982501245,18516210906853,78536526586553,333712398776281,1420364536094093
%N A328494 Constant term in the expansion of (1+x+y+1/x+1/y)^n without assuming commutativity.
%C A328494 Related to A035610 which is the constant term of (x+y+1/x+1/y)^(2n).
%C A328494 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
%H A328494 Andrew Howroyd, <a href="/A328494/b328494.txt">Table of n, a(n) for n = 0..500</a>
%H A328494 Mark Haiman, <a href="https://doi.org/10.1006/eujc.1993.1036">Non-commutative rational power series and algebraic generating functions</a>, European Journal of Combinatorics, 14(4):335-9 (1993).
%H A328494 Robin Hankin, <a href="/A328494/a328494.pdf">Discussion of this and similar sequences</a>
%H A328494 Pakawut Jiradilok and Supanat Kamtue, <a href="https://arxiv.org/abs/2107.09876">Transportation Distance between Probability Measures on the Infinite Regular Tree</a>, arXiv:2107.09876 [math.CO], 2021.
%F A328494 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A035610(k). - _Andrew Howroyd_, Oct 25 2019
%p A328494 h := n -> GAMMA(n+1/2)/GAMMA(n+2)*hypergeom([2, 1-n], [n+2], -3):
%p A328494 a := n -> 3-(-3)^n-5^n+(1/sqrt(Pi))*add(12^(k+1)*binomial(n, 2*k)*h(k), k=1..n/2):
%p A328494 seq(simplify(a(n)), n=0..26); # _Peter Luschny_, Oct 25 2019
%o A328494 (R)
%o A328494 library("freealg")
%o A328494 g <- function(p,string){constant(as.freealg(string)^p)} sapply(0:7,g,"1+x+y+X+Y")
%o A328494 (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
%Y A328494 Cf. A035610, A201805.
%K A328494 nonn
%O A328494 0,3
%A A328494 _Robin Hankin_, Oct 16 2019
%E A328494 Terms a(8) and beyond from _Andrew Howroyd_, Oct 25 2019