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A136519
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a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients.
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1
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1, 3, 15, 156, 4556, 417384, 128004240, 136874853504, 523288667468832, 7257782720507161152, 368292386875012729754240, 68761030015590030510485191680, 47447175348985315294381264871833600
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OFFSET
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0,2
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COMMENTS
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a(n) = [x^n] (1 + x + x^2)^(2^n+1), the coefficient of x^n in (1 + x + x^2)^(2^n+1).
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LINKS
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Table of n, a(n) for n=0..12.
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FORMULA
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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!.
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EXAMPLE
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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.
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PROG
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(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)
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CROSSREFS
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Cf. A027907, A136518.
Sequence in context: A228901 A195226 A264558 * A102556 A016065 A005016
Adjacent sequences: A136516 A136517 A136518 * A136520 A136521 A136522
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 02 2008
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STATUS
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approved
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