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A136519 a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients. 1
1, 3, 15, 156, 4556, 417384, 128004240, 136874853504, 523288667468832, 7257782720507161152, 368292386875012729754240, 68761030015590030510485191680, 47447175348985315294381264871833600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..12.

FORMULA

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!.

EXAMPLE

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

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! +...

This is a special case of the more general statement:

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

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

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

PROG

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

(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)

CROSSREFS

Cf. A027907, A136518.

Sequence in context: A228901 A195226 A264558 * A102556 A016065 A005016

Adjacent sequences:  A136516 A136517 A136518 * A136520 A136521 A136522

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 02 2008

STATUS

approved

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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)