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 A136519 a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients. 1

%I #4 Jun 13 2015 10:24:59

%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.

%C 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 + 2^(2n)*x^2) * log(1 + 2^n*x + 2^(2n)*x^2)^n / n!.

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

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

%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)

%Y Cf. A027907, A136518.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 02 2008

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Last modified June 5 03:05 EDT 2023. Contains 363130 sequences. (Running on oeis4.)