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A136516
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a(n) = (2^n+1)^n.
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6
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1, 3, 25, 729, 83521, 39135393, 75418890625, 594467302491009, 19031147999601100801, 2460686496619787545743873, 1280084544196357822418212890625, 2672769719437237714909813214827010049
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! for all q, m, b.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..45
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FORMULA
| E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(2^n*x) * x^n/n!.
O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 - 2^n*x)^(n+1) = Sum_{n>=0} (2^n+1)^n*x^n. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
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EXAMPLE
| A(x) = 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +... + (2^n+1)^n*x^n/n! +...
A(x) = exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+ 2^(n^2)*exp(2^n*x)*x^n/n! +...
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) = exp(x), q=2, m=1, b=1.
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MATHEMATICA
| Table[(2^n+1)^n, {n, 0, 16}] (*From Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
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PROG
| (PARI) a(n)=polcoeff(sum(k=0, n, 2^(k^2)*exp(2^k*x)*x^k/k!), n)
(PARI) {a(n)=polcoeff(sum(k=0, n, 2^(k^2)*x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
(MAGMA) [(2^n+1)^n: n in [0..45]]; // Vincenzo Librandi, Apr 21 2011
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CROSSREFS
| Cf. A055601.
Sequence in context: A131310 A127231 A062411 * A002021 A012764 A101733
Adjacent sequences: A136513 A136514 A136515 * A136517 A136518 A136519
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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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