A381817 Expansion of (1/x) * Series_Reversion( x * (1-x) / C(x) ), where C(x) is the g.f. of A000108.
1, 2, 8, 41, 239, 1507, 10016, 69123, 490676, 3560150, 26285896, 196862679, 1491921261, 11420072162, 88166571504, 685724643699, 5367842153463, 42259058503891, 334373741310812, 2657683458672907, 21209720057079565, 169886023881795700, 1365290865904393560
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*2*x/(1-sqrt(1-4*x)))/x)
Formula
G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(2*n-k,n-k)/(n+2*k+1).
D-finite with recurrence 270*n*(n-1)*(2*n+1)*(4886806261*n -12359738163)*(n+1)*a(n) +36*n*(n-1)*(73302093915*n^3 -4013759132354*n^2 +11228589268975*n -4731576382254)*a(n-1) -6*(n-1)*(78948725805818*n^4 -721014042837927*n^3 +2114039183987386*n^2 -2373558292742247*n +834825525358878)*a(n-2) +(3703469060597227*n^5 -40768871113864973*n^4 +173554734639707111*n^3 -360669855974794759*n^2 +370762762031723274*n -153683482287306096)*a(n-3) +6*(-2284895393144753*n^5 +28245013068548213*n^4 -138588666805096327*n^3 +341806596235129383*n^2 -433338949590369664*n +232825263110939100)*a(n-4) +10*(5*n-22)*(5*n-21) *(5*n-19)*(5*n-18)*(1032930487477*n -4077934418263)*a(n-5)=0. - R. J. Mathar, Mar 10 2025