OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (n+3)/(n+2) * Sum_{k=1..n/2} C(n+k+1,k)*C(n-k-1,k-1)*(n+2*k)/(n+k+1) , n>0.
Recurrence: 5*(n-2)*n*(n+1)^2*(n+2)^2*(481*n^2 - 1003*n + 540)*a(n) = 4*n*(n+1)^2*(n+3)*(962*n^4 - 3449*n^3 + 3262*n^2 - 784*n - 180)*a(n-1) + 4*(n-1)*n*(n+2)*(n+3)*(3848*n^4 - 11872*n^3 + 12073*n^2 - 3572*n - 180)*a(n-2) - 2*(n-2)*(n-1)*(n+1)*(n+2)*(n+3)*(2*n-5)*(481*n^2 - 41*n + 18)*a(n-3). - Vaclav Kotesovec, Feb 19 2015
a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.40869819984215108586... is the root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0, and c = 0.838651324525827608604668464... is the root of the equation 169 + 157184*c^2 - 275872*c^4 + 74000*c^6 = 0. - Vaclav Kotesovec, Feb 21 2015
MAPLE
a:=n->(n+3)/(n+2)*sum(binomial(n+k+1, k)*binomial(n-k-1, k-1)*(n+2*k)/(n+k+1), k=1..trunc(n/2)): (1, seq(a(n), n=1..30));
MATHEMATICA
Flatten[{1, Table[(n+3)/(n+2)*Sum[Binomial[n+k+1, k]*Binomial[n-k-1, k-1]*(n+2k)/(n+k+1), {k, Floor[n/2]}], {n, 20}]}] (* Vaclav Kotesovec, Feb 19 2015 *)
PROG
(PARI) a(n) = if (n==0, 1, (n+3)*sum(k=1, n\2, binomial(n+k+1, k)*binomial(n-k-1, k-1)*(n+2*k)/(n+k+1))/(n+2)); \\ Michel Marcus, Mar 03 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael D. Weiner, Feb 16 2015
EXTENSIONS
Definition clarified by Michael D. Weiner, Mar 09 2015
STATUS
approved