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A136465
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Row 0 of square array A136462: a(n) = C(2^(n-1), n) for n>=0.
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5
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1, 1, 1, 4, 70, 4368, 906192, 621216192, 1429702652400, 11288510714272000, 312268282598377321216, 30813235422145714150738944, 11005261717918037175659349191168
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n) is found in row n, column 0, of triangle A136467 for n>=0.
For n > 0, number of increasing integer sequences 1 <= a_1 < ... < a_n <= 2^(n-1). [From Charles R Greathouse IV (charles.greathouse(AT)case.edu), Aug 08 2010]
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FORMULA
| a(n) = [x^n] Sum_{i>=0} (1/2)^i * log(1 + 2^i*x)^i/i!.
G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n/(n!*2^n) ). [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 26 2010]
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EXAMPLE
| Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 26 2010: (Start)
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 70*x^4 + 4368*x^5 +...
A(x) = 1 + log(1+2*x)/2 + log(1 + 2^2*x)^2/(2!*2^2) + log(1 + 2^3*x)^3/(3!*2^3) + log(1 + 2^4*x)^4/(4!*2^4) +... (End)
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PROG
| (PARI) {a(n)=binomial(2^(n-1), n)} (PARI) /* a(n) = Coefficient of x^k in series: */ {a(n)=polcoeff(sum(i=0, n, (1/2)^i*log(1+2^i*x +x*O(x^n))^i/i!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, log(1+2^m*x+x*O(x^n))^m/(m!*2^m)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 26 2010]
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CROSSREFS
| Cf. A136462; other rows: A014070, A136466, A101346; A136467.
Sequence in context: A065573 A101841 A061609 * A184576 A162135 A047939
Adjacent sequences: A136462 A136463 A136464 * A136466 A136467 A136468
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Dec 31 2007
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