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a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(n+1,n-3*k).
(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x^3)^2))/x)
Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x^3)^2 ) ).
Cf. A198951.
<a href="/index/Res#revert">Index entries for reversions of series</a>
1, 1, 1, 3, 11, 31, 84, 261, 865, 2815, 9131, 30339, 102681, 349376, 1193993, 4111947, 14263137, 49720513, 174040102, 611770893, 2158954383, 7645030641, 27153898487, 96719683491, 345414958227, 1236555046701, 4436564115556, 15950469680836, 57455730349552
(PARI) a(n, s=3, t=1, 2, u=21) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);
allocated for Seiichi Manyama
Expansion of (1/x) * Series_Reversion( x / (1+x) * (1+x^3)^2 ).
1, 1, 1, 3, 11, 31, 84, 261, 865, 2815, 9131, 30339, 102681, 349376, 1193993, 4111947, 14263137, 49720513, 174040102, 611770893, 2158954383, 7645030641, 27153898487, 96719683491
0,4
(PARI) a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);
allocated
nonn
Seiichi Manyama, Jan 23 2024
approved
editing
allocated for Seiichi Manyama
allocated
approved