The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!. 2

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 29 04:26 EDT 2024. Contains 372921 sequences. (Running on oeis4.)