login
a(n) = 1 + Sum_{k=0..n-1} binomial(n+3,k+4) * a(k).
1

%I #5 Apr 07 2022 09:22:35

%S 1,2,8,36,170,865,4742,27757,172375,1130865,7809057,56572404,

%T 428710587,3389749264,27901667938,238599540142,2115876327408,

%U 19425465343555,184355895494512,1806122902809371,18242807108024625,189750478368293523,2030261803964224359,22323607721661782198

%N a(n) = 1 + Sum_{k=0..n-1} binomial(n+3,k+4) * a(k).

%F G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * A(x/(1 - x)) / (1 - x)^5.

%F a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(n+2,k+3) * a(k).

%t a[n_] := a[n] = 1 + Sum[Binomial[n + 3, k + 4] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]

%t nmax = 23; A[_] = 0; Do[A[x_] = 1/(1 - x) + x A[x/(1 - x)]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A000110, A032346, A032347, A045499, A045500, A186021, A352861.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 06 2022