A196148 Antidiagonal sums of square array A111910.

1, 2, 7, 30, 146, 772, 4331, 25398, 154158, 961820, 6137734, 39909740, 263665252, 1765815560, 11966535091, 81937361702, 566185489878, 3944202596652, 27676632525362, 195481707009220, 1388890568962556

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

Cf. A111910.

Cf. A174119.

Sequence in context: A368932 A369160 A243632 * A193464 A166990 A059578

Adjacent sequences: A196145 A196146 A196147 * A196149 A196150 A196151

Keyword

nonn,easy

Author

Peter Bala, Oct 13 2011

Status

approved