A363390 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / (k*x^k) ).
1, 2, 9, 60, 436, 3462, 28810, 248606, 2202772, 19929336, 183331451, 1709642222, 16125333248, 153564283602, 1474528190435, 14260019116712, 138772479615509, 1357948477513772, 13353454737592303, 131889469476063586, 1307802326452419584, 13014461023695752740
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Keywords
Programs
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Mathematica
nmax = 22; A[] = 0; Do[A[x] = x Exp[2 Sum[(-1)^(k + 1) A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest 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[(-1)^(k/d + 1) d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 22}]
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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, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p) \\ Andrew Howroyd, May 30 2023