OFFSET
0,2
FORMULA
a(n) = sum(j=0, n, C(n,j)^2 * A005802(j)).
a(n) = sum(j=0, n, C(n,j)^2 * (1/((j+1)^2 (j+2))) * sum(i=0, j, C(2*i,i) * C(j+1,i+i) * C(j+2,i+1))) where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)^3*(3*n+1)*a(n) = 2*(30*n^4 + 67*n^3 + 29*n^2 - 10*n - 8)*a(n-1) - 64*(n-1)^2*n*(3*n+4)*a(n-2). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 2^(4*n+5) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Feb 18 2015
MATHEMATICA
Table[Sum[ Binomial[n, j]^2*(1/((j + 1)^2*(j + 2)))* Sum[Binomial[2*i, i]*Binomial[j + 1, i + 1]* Binomial[j + 2, i + 1], {i, 0, j}], {j, 0, n}], {n, 0, 20}]
PROG
(PARI) /* using formula given; this gives fractions! */
C=binomial;
a(n)=sum(j=0, n, C(n, j)^2 * (1/((j+1)^2*(j+2))) * sum(i=0, j, C(2*i, i)*C(j+1, i+i)*C(j+2, i+1)));
\\ Joerg Arndt, Feb 18 2015
(PARI) /* Using a(n) = sum(j=0, n, C(n, j)^2 * A005802(j)). */
f(n)= 2 * sum(k=0, n, binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2+2*k+1-n-2*k*n)/((k+1)^2 * (k+2) * (n-k+1)));
vector(33, N, my(n=N-1); sum(j=0, n, f(j) * C(n, j)^2 ) )
\\ Joerg Arndt, Feb 18 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric S. Egge, Oct 21 2008
EXTENSIONS
Added more terms, Joerg Arndt, Feb 18 2015
STATUS
approved