A300048 G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^2 + x*A(x)^6.

1, 1, 2, 7, 29, 131, 627, 3124, 16032, 84162, 449828, 2439550, 13391105, 74256824, 415357737, 2340775363, 13278009018, 75753246286, 434392031856, 2502289328542, 14473290097526, 84023214062635, 489424396591995, 2859551104564120, 16754209625090980, 98415932763515679, 579475837597933632, 3419452319373566239, 20219028961691299994

Offset: 0

Author

Paul D. Hanna, Feb 25 2018

Keywords

• nonn

Comments

Compare to: G(x)^3 = 1 + x*G(x) + x*G(x)^2 + x*G(x)^3 holds when G(x) = 1/(1-x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 131*x^5 + 627*x^6 + 3124*x^7 + 16032*x^8 + 84162*x^9 + 449828*x^10 + 2439550*x^11 + 13391105*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 348*x^5 + 1681*x^6 + 8432*x^7 + 43495*x^8 + 229260*x^9 + 1229371*x^10 + ...
A(x)^6 = 1 + 6*x + 27*x^2 + 122*x^3 + 579*x^4 + 2862*x^5 + 14588*x^6 + 76146*x^7 + 405039*x^8 + 2187756*x^9 + 11967426*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 684*x^5 + 3341*x^6 + 16896*x^7 + 87702*x^8 + 464566*x^9 + 2501178*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000108, A106228, A219537.

Programs

PARI

{a(n) = my(A=1); for(i=1,n, A = (1 + x*A + x*A^2 + x*A^6 +x*O(x^n))^(1/3) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ sqrt(1 + sqrt((9 + 40*sqrt(3))/13)) * (9 + 6*sqrt(3) + sqrt(153 + 100*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 3/2) * 3^(n + 3/4)). - Vaclav Kotesovec, Aug 11 2021

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023