A260701 a(n) = (-3*(-1)^n + Sum_{k>=0} A000108(k)*k^n/6^k)/sqrt(3), where A000108 are Catalan numbers.
-1, 2, -1, 5, 20, 197, 2219, 30620, 496565, 9265037, 195535514, 4605925535, 119796721835, 3410051954402, 105449267146859, 3520120318516625, 126168879827914580, 4832661370036811417, 197001989531658791879, 8515772839409988885140, 389080859811496699020425
Offset: 0
Keywords
Examples
For n = 5, Sum_{k>=0} A000108(k)*k^5/6^k = 197*sqrt(3) - 3, so a(5) = 197.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Programs
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Mathematica
Table[(-3 (-1)^n + Sum[CatalanNumber[k] k^n/6^k, {k, 0, Infinity}])/Sqrt[3], {n, 0, 20}] -
PARI
vector(20, n, n--; round((suminf(k=0, binomial(2*k,k)/(k+1)*k^n/6^k) - 3*(-1)^n)/sqrt(3))) \\ Altug Alkan, Nov 16 2015
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PARI
N=20; x='x+O('x^N); Vec(serlaplace(-sqrt(exp(-x)*(-2+3*exp(-x))))) \\ Seiichi Manyama, Oct 21 2021
Formula
Sum_{k >= 0} A000108(k)*k^n/6^k = a(n)*sqrt(3) + 3*(-1)^n.
a(n) ~ sqrt(2) * n^(n-1) / (sqrt(3) * exp(n) * log(3/2)^(n-1/2)). - Vaclav Kotesovec, Nov 17 2015
E.g.f.: -sqrt( exp(-x) * (-2+3*exp(-x)) ). - Seiichi Manyama, Oct 21 2021