OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 1..200 from Vincenzo Librandi)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems.
L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae (1971), 6 (2-3), 256-262.
FORMULA
a(n) = 2 *( Sum_{k=0..floor(n/2)} binomial(n-1, k) * ( binomial(n-1, k) * binomial(2n+1-2k, n+1) + binomial(n-1, k+1)*binomial(2n-2k, n+1)) ).
D-finite with recurrence: n*(n+1)*a(n) = (n+1)*(7*n-4)*a(n-1) + 8*(n-2)^2*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 9*sqrt(3)*2^(3*n-2)/(Pi*n). - Vaclav Kotesovec, Oct 18 2012
G.f.: (2-x)*(1-8*x)^(-1/3)*(x+1)^(-2/3)*hypergeom([1/3, 1/3],[1],27*x^2/(8*x-1)/(x+1)^2) + 3*x*(2*x-1)^2*(1-8*x)^(-4/3)*(x+1)^(-8/3) * hypergeom([4/3, 4/3],[2],27*x^2/(8*x-1)/(x+1)^2) - 2. - Mark van Hoeij, May 14 2013
a(n) = 6*A190917(n) for n >= 1. - R. J. Mathar, Nov 01 2015
MAPLE
a:= proc(n) option remember; `if`(n<2, 1+5*n,
((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
end:
seq(a(n), n=0..21); # Alois P. Heinz, Sep 09 2023
MATHEMATICA
Table[2*(Sum[Binomial[n-1, k]*(Binomial[n-1, k]*Binomial[2n+1-2k, n+1]+Binomial[n-1, k+1]*Binomial[2n-2k, n+1]), {k, 0, Floor[n/2]}]), {n, 1, 20}] (* Vaclav Kotesovec, Oct 18 2012 *)
Table[2 (Binomial[2 n + 1, n + 1] HypergeometricPFQ[{1 - n, 1 - n, 1/2 - n/2, -(n/2)}, {1, -(1/2) - n, -n}, 1] + (n - 1) Binomial[2 n, n + 1] HypergeometricPFQ[{1 - n, 2 - n, 1/2 - n/2, 1 - n/2}, {2, 1/2 - n, -n}, 1]), {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
RecurrenceTable[{n(n+1)*a[n] == (n+1)*(7*n-4)*a[n-1] +8*(n-2)^2*a[n-2], a[1]==6, a[2]==30}, a, {n, 10}] (* Eric W. Weisstein, May 27 2017 *)
PROG
(PARI) a(n)=2*sum(k=0, n\2, binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1)))
(Magma) [2*(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Nov 24 2018
(Sage) [2*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1)) for k in range(1+floor(n/2))) for n in (1..25)] # G. C. Greubel, Nov 24 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Aug 04 2005
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Sep 09 2023
STATUS
approved