OFFSET
0,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1000
Anthony James Wood, Nonequilibrium steady states from a random-walk perspective, Ph. D. Thesis, The University of Edinburgh (Scotland, UK 2019).
Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, Renyi entropy of the totally asymmetric exclusion process, arXiv:1708.00303 [cond-mat.stat-mech], 2017.
Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, Combinatorial mappings of exclusion processes, arXiv:1908.00942 [cond-mat.stat-mech], 2019.
FORMULA
a(n) = Sum_{k = 0..n} S(n-k,k) where S(n,k) = (n+k+1)!*(2*n+2*k+1)!/((n+1)!*(k+1)!*(2*n+1)!*(2*k+1)!).
From Vaclav Kotesovec, Dec 16 2017: (Start)
a(n) ~ 2^(3*n+3) / (sqrt(3*Pi) * n^(5/2)).
Recurrence: (n+2)*(2*n+3)*a(n) = 2*(7*n^2 + 7*n + 1)*a(n-1) + 8*(n-1)*(2*n-1)*a(n-2). (End)
a(n) = hypergeometric3F2([-n, -n-1/2, -n-1], [3/2, 2], -1). - G. C. Greubel, Feb 11 2021
Let E(x) = Sum_{n >= 0} x^n/((n+1)!*(2*n+1)!). Then E(x)^2 = 1 + 2*x/(2!*3!) + 7*x^2/(3!*5!) + 30*x^3/(4!*7!) + ... + a(n)*x^n/((n+1)!*(2*n+1)!) + ... is a generating function for the sequence. - Peter Bala, Sep 20 2021
MATHEMATICA
Table[Sum[(n+1)! * (2*n+1)! / ((n-k+1)! * (k+1)! * (2*n-2*k+1)! * (2*k+1)!), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2017 *)
Table[HypergeometricPFQ[{-n, -n-1/2, -n-1}, {3/2, 2}, -1], {n, 0, 25}] (* G. C. Greubel, Feb 11 2021 *)
PROG
(PARI) S(n, k) = (n+k+1)!*(2*n+2*k+1)!/((n+1)!*(k+1)!*(2*n+1)!*(2*k+1)!);
a(n) = sum(k = 0, n, S(n-k, k)); \\ Michel Marcus, Dec 16 2017
(Sage) [hypergeometric([-n, -n-1/2, -n-1], [3/2, 2], -1).simplify_hypergeometric() for n in (0..25)] # G. C. Greubel, Feb 11 2021
(Magma) [(&+[(n-j+1)*Binomial(n+1, j)*Binomial(2*n+4, 2*j+2)/((n+1)*(n+2)*(2*n+3)): j in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 11 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Oct 13 2011
STATUS
approved