A371888 - OEIS

A371888

G.f. A(x) satisfies A(x) = 1 - x/A(x) * (1 - A(x) - A(x)^2).

1, 1, 2, 3, 3, 1, -2, -1, 10, 25, 12, -65, -151, -7, 588, 1083, -437, -5247, -7732, 7943, 47503, 53793, -105312, -430117, -343042, 1249801, 3866558, 1730019, -13996095, -34243895, -1947202, 150962375, 296101866, -121857183, -1582561868, -2468098041, 2529520767

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..36.

FORMULA

a(n) = (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n-2*k,n-k-1) for n > 0.

a(n) = (1/2) * Sum_{k=0..n} 4^k * binomial(k/2+1/2,k) * binomial(n-1,n-k)/(k+1) for n > 0.

G.f.: A(x) = 2*x/(1+x - sqrt(1-2*x+5*x^2)).

D-finite with recurrence n*a(n) +3*(-n+1)*a(n-1) +(7*n-18)*a(n-2) +5*(-n+3)*a(n-3)=0. - R. J. Mathar, Apr 22 2024

PROG

(PARI) a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(n-2*k, n-k-1))/n);