A361304 - OEIS

%I #7 Mar 09 2023 09:00:46

%S 1,2,18,124,930,7146,55804,441312,3521898,28307510,228820086,

%T 1858240956,15149110912,123905220292,1016261712240,8355494725376,

%U 68842600563918,568266625104498,4698576694639306,38906632384471820,322596353513983626,2678048134387075560

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

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

%F (1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.

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

%F (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.

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

%e 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 + ...

%o (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* Using series reversion (faster) */

%o {a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^4 )); polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A361306, A214372.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 08 2023