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A136518
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a(n) = A027907(2^n, n), where A027907 = triangle of trinomial coefficients.
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1
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1, 2, 10, 112, 3620, 360096, 116950848, 129755798400, 507413158135840, 7132358041777380352, 364730093112968976177664, 68393665694364347188157159424, 47308574208170527265149009962117120
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) = [x^n] (1 + x + x^2)^(2^n), the coefficient of x^n in (1 + x + x^2)^(2^n).
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FORMULA
| O.g.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x + 2^(2n)*x^2)^n / n!.
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EXAMPLE
| A(x) = 1 + 2x + 10x^2 + 112x^3 + 3620x^4 + 360096x^5 + ...
A(x) = 1 + log(1+2x+4x^2) + log(1+4x+16x^2)^2/2! + log(1+8x+64x^2)^3/3! +...
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=0.
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PROG
| (PARI) {a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n), n)} (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)}
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CROSSREFS
| Cf. A027907; A136519.
Sequence in context: A181445 A062499 A049505 * A168369 A005613 A005616
Adjacent sequences: A136515 A136516 A136517 * A136519 A136520 A136521
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 02 2008
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