OFFSET
0,3
FORMULA
O.g.f.: ((-3*x^2 + 5*x + 4)*sqrt(x^2 - 6*x + 1) - 3*x^3 + 2*x^2 + 5*x)/(sqrt(x^2 - 6*x + 1)*(6*x^2 + 10*x + 4)).
a(n) ~ sqrt(9*sqrt(2) - 8) * (1 + sqrt(2))^(2*n) / (14*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021
D-finite with recurrence -2*(n-1)*(56*n^2-328*n+467)*a(n) +(392*n^3-3024*n^2+7501*n-5979)*a(n-1) +(1400*n^3-10328*n^2+24
187*n-17824)*a(n-2) +(728*n^3-5216*n^2+11919*n-8571)*a(n-3) -3*(n-4)*(56*n^2-216*n+195)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
D-finite with recurrence 10*(-n+1)*a(n) +(31*n-53)*a(n-1) +(145*n-274)*a(n-2) +2*(47*n-69)*a(n-3) +2*(-32*n+179)*a(n-4) +3*(-15*n+77)*a(n-5) +9*(n-6)*a(n-6)=0. - R. J. Mathar, Mar 06 2022
EXAMPLE
a(3) = (-1)^3*(1 - 7 + 12 - 18) = 12.
a(4) = (-1)^4*(1 - 15 + 32 - 56 + 88) = 50.
PROG
(PARI) G(n, k) = if (k==0, 1, sum(j=1, n, binomial(n, j)*binomial(k+j-2, j-1))); \\ A111516
a(n) = sum(k=0, n, (-1)^(n+k)*G(n, k)); \\ Michel Marcus, Feb 14 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Feb 14 2021
STATUS
approved