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%I #9 May 31 2022 02:39:27
%S 1,1,2,7,32,179,1184,8977,76391,719132,7405261,82654011,992533974,
%T 12744345310,174073918884,2518084939316,38429337167618,
%U 616676966998463,10374679318111371,182506045254212184,3349265281648290030,63984975864984809787,1270096455615572678617
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^3) / (1 - x).
%H Seiichi Manyama, <a href="/A351813/b351813.txt">Table of n, a(n) for n = 0..486</a>
%F a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+2*k-1,n-k-1) * a(k).
%t nmax = 22; A[_] = 0; Do[A[x_] = 1 + x A[x/(1 - x)^3]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 2 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]
%Y Cf. A000110, A125273.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 19 2022