A383935 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(1/3).
1, 3, 12, 54, 261, 1323, 6930, 37152, 202554, 1118286, 6233760, 35014356, 197881866, 1123990182, 6411554028, 36705925656, 210797967321, 1213895891835, 7007131607220, 40534622188830, 234931402041525, 1363961443750155, 7931187074571930, 46183636475060760
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025 -
Mathematica
CoefficientList[Series[1/((1-3*x)*(1-6*x))^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *) -
PARI
a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-1/3, k)*binomial(-1/3, n-k));
Formula
G.f.: B(x)^(1/3), where B(x) is the g.f. of A016137.
a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-1/3,k) * binomial(-1/3,n-k).
a(n) ~ 2^(n + 1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence: (-9*n-3)*a(n)+(18*n-6)*a(n-1)+(n+1)*a(n+1) = 0. - Georg Fischer, Aug 29 2025