A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).
1, 3, 16, 106, 786, 6244, 51964, 447201, 3947306, 35538668, 325098696, 3013060258, 28232408848, 267003169668, 2545341982728, 24433290332007, 235967943943224, 2291147902820524, 22352525061549604, 219006814853751540, 2154083325737401740
Offset: 0
Keywords
Programs
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Mathematica
CoefficientList[InverseSeries[Series[x*((1-x)^3 - x^2), {x, 0, 30}], x]/x, x](* Vaclav Kotesovec, Jan 29 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^2))/x) -
PARI
a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n+k+2,n-2*k).
a(n) ~ sqrt((60 + (220324 - 42734*sqrt(2))^(1/3) + (220324 + 42734*sqrt(2))^(1/3)) / (138*Pi)) * (((4/23)*(22 + 3*(293 - 92*sqrt(2))^(1/3) + 3*(293 + 92*sqrt(2))^(1/3)))^n / n^(3/2)). - Vaclav Kotesovec, Jan 29 2024