A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.

1, 2, 18, 124, 930, 7146, 55804, 441312, 3521898, 28307510, 228820086, 1858240956, 15149110912, 123905220292, 1016261712240, 8355494725376, 68842600563918, 568266625104498, 4698576694639306, 38906632384471820, 322596353513983626, 2678048134387075560

Offset

0,2

Links

Table of n, a(n) for n=0..21.

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)^(4*n) / n!.

(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^4).

(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)^(4*n) / n! ) is the g.f. of A361306.

(4) a(n) = (n+1) * A361306(n+1) for n >= 0.

Example

G.f.: A(x) = 1 + 2*x + 18*x^2 + 124*x^3 + 930*x^4 + 7146*x^5 + 55804*x^6 + 441312*x^7 + 3521898*x^8 + 28307510*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)))^(4*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)))^4 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A361306, A214372.

Sequence in context: A052653 A342124 A289830 * A358952 A060589 A325275

Adjacent sequences: A361301 A361302 A361303 * A361305 A361306 A361307

Keyword

nonn

Author

Paul D. Hanna, Mar 08 2023

Status

approved