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
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.
Programs
-
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
-
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 *) -
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
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