OFFSET
0,2
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^3).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ) is the g.f. of A361305.
(4) a(n) = (n+1) * A361305(n+1) for n >= 0.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 15*x^2 + 92*x^3 + 615*x^4 + 4200*x^5 + 29190*x^6 + 205416*x^7 + 1458909*x^8 + 10436030*x^9 + ...
PROG
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(3*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^3 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 08 2023
STATUS
approved