OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)^2 + 2*x^2*A(x)^4.
(2) A(x) = (1 - x*A(x) - sqrt(1 - 2*x*A(x) - 7*x^2*A(x)^2)) / (4*x^2*A(x)^2).
(3) A(x) = sqrt( (1/x) * Series_Reversion( x/(1 + x + 2*x^2)^2 ) ).
(4) A(x) = (1/x) * Series_Reversion(x/G(x)) where G(x) = (1-x - sqrt(1 - 2*x - 7*x^2))/(4*x^2) is the g.f. of A025235.
(5) a(n) = (1/(2*n+1)) * Sum_{k=[3*n/2]..2*n+1} 2^(2*n+1-k) * binomial(2*n+1,k) * binomial(k,2*k-2-3*n).
a(n) ~ 2^(2*n) * 7^(2*n + 3/2) / (sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, May 17 2026
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 86*x^4 + 462*x^5 + 2618*x^6 + 15353*x^7 + 92486*x^8 + 568718*x^9 + ...
MATHEMATICA
Table[Sum[2^(2*n + 1 - k) * Binomial[2*n + 1, k] * Binomial[k, 2*k - 2 - 3*n], {k, Floor[3*n/2], 2*n + 1}]/(2*n+1), {n, 0, 25}] (* Vaclav Kotesovec, May 17 2026 *)
PROG
(PARI) {a(n) = (1/(2*n+1))*sum(k=floor(3*n/2+1), 2*n+1, 2^(2*n+1-k) * binomial(2*n+1, k) * binomial(k, 2*k-2-3*n) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = sqrt( 1/x*serreverse(x/(1 + x + 2*x^2)^2 +x^2*O(x^n)) )); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, May 17 2026
STATUS
approved
