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A126694
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Expansion of g.f.: 1/(1 - 7*x*c(x)), where c(x) is the g.f. for A000108.
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9
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1, 7, 56, 455, 3710, 30282, 247254, 2019087, 16488710, 134656130, 1099686056, 8980749862, 73342721956, 598965319960, 4891549246290, 39947649057855, 326239122661830, 2664286127154330, 21758336553841440, 177693081299126610
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OFFSET
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0,2
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COMMENTS
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The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.
Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from c(x) = 1/(1 - x*c(x)) == 1/(1 - 7*x*c(x)) (mod 2). - Peter Bala, Jul 24 2016
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LINKS
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FORMULA
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a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.
a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.
a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.
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MATHEMATICA
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CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, May 05 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // G. C. Greubel, May 05 2019
(Sage) (2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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