cp's OEIS Frontend

A136518 a(n) = A027907(2^n, n), where A027907 = triangle of trinomial coefficients.

%I A136518 #12 Jul 28 2023 04:58:04
%S A136518 1,2,10,112,3620,360096,116950848,129755798400,507413158135840,
%T A136518 7132358041777380352,364730093112968976177664,
%U A136518 68393665694364347188157159424,47308574208170527265149009962117120
%N A136518 a(n) = A027907(2^n, n), where A027907 = triangle of trinomial coefficients.
%C A136518 This is a special case of the more general statement:
%C A136518   Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
%C A136518   Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)
%C A136518 where F(x) = 1+x+x^2, q=2, m=1, b=0.
%H A136518 G. C. Greubel, <a href="/A136518/b136518.txt">Table of n, a(n) for n = 0..59</a>
%F A136518 a(n) = [x^n] (1 + x + x^2)^(2^n), the coefficient of x^n in (1 + x + x^2)^(2^n).
%F A136518 O.g.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x + 4^n*x^2)^n / n!.
%e A136518 A(x) = 1 + 2*x + 10*x^2 + 112*x^3 + 3620*x^4 + 360096*x^5 + ...
%e A136518 A(x) = 1 + log(1 +2*x +4*x^2) + log(1 +4*x +16*x^2)^2/2! + log(1 +8*x +64*x^2)^3/3! + ...
%t A136518 With[{m=40, f= 1 +2^j*x +4^j*x^2}, CoefficientList[Series[ Sum[Log[f]^j/j!, {j,0,m+1}], {x,0,m}], x]] (* _G. C. Greubel_, Jul 27 2023 *)
%o A136518 (PARI) a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n),n)
%o A136518 (PARI) /* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,log(1+2^i*x+2^(2*i)*x^2 +x*O(x^n))^i/i!),n)
%o A136518 (Magma)
%o A136518 m:=40;
%o A136518 gf:= func< x | (&+[Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >;
%o A136518 R<x>:=PowerSeriesRing(Rationals(), m);
%o A136518 Coefficients(R!( gf(x) )); // _G. C. Greubel_, Jul 27 2023
%o A136518 (SageMath)
%o A136518 m=40
%o A136518 def f(x): return sum( log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) )
%o A136518 def A136518_list(prec):
%o A136518     P.<x> = PowerSeriesRing(QQ, prec)
%o A136518     return P( f(x) ).list()
%o A136518 A136518_list(m) # _G. C. Greubel_, Jul 27 2023
%Y A136518 Cf. A027907, A136519.
%K A136518 nonn
%O A136518 0,2
%A A136518 _Paul D. Hanna_, Jan 02 2008