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A046714
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Convolution of A000108 (Catalan) with A000351 (powers of 5).
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5
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1, 6, 32, 165, 839, 4237, 21317, 107014, 536500, 2687362, 13453606, 67326816, 336842092, 1684953360, 8427441240, 42146901045, 210769862895, 1053978959265, 5270372435025, 26353629438315, 131774711311995
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} A000108(k)*5^(n-k).
a(n) = 5*a(n-1) + C(n), a(0) = 1.
G.f.: c(x)/(1-5*x), where c(x) = g.f. for Catalan numbers A000108.
Homogeneous recursion: a(n) = (3*(3*n+1)/(n+1))*a(n-1) - (10*(2*n-1)/(n+1))*a(n-2), a(-1) := 0, a(0)=1, n >= 1.
Hypergeometric 2F1 form: 2*a(n) = 5^(n+1) - binomial(2*(n+1), n+1) * hypergeom([ -n-1, 1 ], [ 1/2 ], -1/4).
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x, 0, 40}], x] (* G. C. Greubel, Jul 28 2024 *)
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PROG
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(Magma)
[n le 1 select 1 else 5*Self(n-1) + Catalan(n-1): n in [1..40]]; // G. C. Greubel, Jul 28 2024
(SageMath)
@CachedFunction
def A046714(n): return 1 if n==1 else 5*A046714(n-1) + catalan_number(n-1)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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