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A363389 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} A(x^k)^2 / (k*x^k) ). 1

%I #7 May 30 2023 19:56:01

%S 1,2,11,72,545,4432,38081,339266,3107841,29080910,276786032,

%T 2671136262,26076724707,257061506994,2555287226253,25584395476368,

%U 257780104545994,2611791146130284,26593326491738879,271972643143865548,2792566207778712513,28776796478486084250

%N G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

%t nmax = 22; A[_] = 0; Do[A[x_] = x Exp[2 Sum[A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

%t a[1] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (2/(n - 1)) Sum[Sum[d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 22}]

%o (PARI) seq(n)=my(p=x+O(x^2)); for(n=2, n, my(m=serprec(p,x)-1); p = x*exp(2*sum(k=1, m, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p) \\ _Andrew Howroyd_, May 30 2023

%Y Cf. A000151, A005750, A363390.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, May 30 2023

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Last modified April 14 11:07 EDT 2024. Contains 371657 sequences. (Running on oeis4.)