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 A317875 Number of achiral free pure multifunctions with n unlabeled leaves. 13
 1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 FORMULA a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d). From Ilya Gutkovskiy, Apr 30 2019: (Start) G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k). G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End) EXAMPLE The first 4 terms count the following multifunctions. o, o[o], o[o,o], o[o[o]], o[o][o], o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o]. MATHEMATICA a[n_]:=If[n==1, 1, Sum[a[n-k]*Sum[a[d], {d, Divisors[k]}], {k, n-1}]]; Array[a, 12] PROG (PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018 (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018 CROSSREFS Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853. Cf. A317876, A317877, A317878, A317879, A317880, A317881. Cf. A317882, A317883, A317884, A317885. Sequence in context: A257641 A048119 A304823 * A056333 A148948 A148949 Adjacent sequences: A317872 A317873 A317874 * A317876 A317877 A317878 KEYWORD nonn AUTHOR Gus Wiseman, Aug 09 2018 STATUS approved

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Last modified June 22 20:54 EDT 2024. Contains 373608 sequences. (Running on oeis4.)