A212029 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.

1, 1, 3, 21, 190, 2112, 26922, 382110, 5920788, 98862273, 1762572957, 33325846461, 664774457583, 13932829786025, 305788481726799, 7008171327166869, 167321925537782445, 4153009604547937170, 106963758805117459392, 2854029374011293902121, 78773444214057182702790

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

From Seiichi Manyama, Mar 01 2025: (Start)

Let a(n,k) = [x^n] A(x)^k.

a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,3*j). (End)

Example

G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 190*x^4 + 2112*x^5 + 26922*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 732*x^4 + 7944*x^5 + 99156*x^6 +..
A(x*A(x)^3) = 1 + x + 6*x^2 + 51*x^3 + 560*x^4 + 7155*x^5 + 102495*x^6 +...
A(x*A(x)^3)^3 = 1 + 3*x + 21*x^2 + 190*x^3 + 2112*x^4 + 26922*x^5 +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025

Crossrefs

Cf. A143508, A143501, A212028, A381570.

Sequence in context: A256625 A057616 A212072 * A219535 A292361 A369783

Adjacent sequences: A212026 A212027 A212028 * A212030 A212031 A212032

Keyword

nonn,changed

Author

Paul D. Hanna, Apr 27 2012

Status

approved