A136519 - OEIS

%I #11 Jul 30 2023 02:12:55

%S 1,3,15,156,4556,417384,128004240,136874853504,523288667468832,

%T 7257782720507161152,368292386875012729754240,

%U 68761030015590030510485191680,47447175348985315294381264871833600

%N a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients.

%H G. C. Greubel, <a href="/A136519/b136519.txt">Table of n, a(n) for n = 0..59</a>

%F a(n) = [x^n] (1 + x + x^2)^(2^n+1), the coefficient of x^n in (1 + x + x^2)^(2^n+1).

%F O.g.f.: A(x) = Sum_{n>=0} (1 + 2^n*x + 4^n*x^2) * log(1 + 2^n*x + 4^n*x^2)^n / n!.

%e A(x) = 1 + 3*x + 15*x^2 + 156*x^3 + 4556*x^4 + 417384*x^5 + ...

%e A(x) = (1 +x +x^2) + (1 +2*x +4*x^2)*log(1 +2*x +4*x^2) + (1 +4*x +16*x^2)*log(1 +4*x +16*x^2)^2/2! + (1 +8*x +64*x^2)*log(1 +8*x +64*x^2)^3/3! + (1 +16*x +256*x^2)*log(1 +16*x +256*x^2)^4/4! + ...

%e This is a special case of the more general statement:

%e   Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =

%e   Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)

%e   where F(x) = 1+x+x^2, q=2, m=1, b=1.

%t With[{m=40, f= 1 +2^j*x +4^j*x^2}, CoefficientList[Series[ Sum[f*Log[f]^j/j!, {j,0,m+1}], {x,0,m}], x]] (* _G. C. Greubel_, Jul 27 2023 *)

%o (PARI) a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n+1),n)

%o (PARI) /* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,(1+2^i*x+2^(2*i)*x^2)*log(1+2^i*x+2^(2*i)*x^2 +x*O(x^n))^i/i!),n)

%o (Magma)

%o m:=40; // gf of A136519

%o gf:= func< x | (&+[(1 +2^j*x +4^j*x^2)*Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o Coefficients(R!( gf(x) )); // _G. C. Greubel_, Jul 27 2023

%o (SageMath)

%o m=40

%o def f(x): return sum( (1 + 2^j*x + 4^j*x^2)*log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) )

%o def A136519_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( f(x) ).list()

%o A136519_list(m) # _G. C. Greubel_, Jul 27 2023

%Y Cf. A027907, A136518.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 02 2008