A345973 G.f.: x + x^2 / Product_{n>=1} (1 - a(n)*x^n).
1, 1, 1, 2, 3, 6, 10, 20, 36, 73, 138, 281, 549, 1136, 2263, 4705, 9553, 20015, 41096, 86643, 179638, 380701, 795892, 1693003, 3562217, 7612680, 16099538, 34505797, 73345831, 157678081, 336419942, 725236780, 1552662599, 3354979195, 7205601904, 15600414855, 33594465666
Offset: 1
Keywords
Links
- David Callan, A Combinatorial Interpretation for Sequence A345973 in OEIS, arXiv:2108.04969 [math.CO], 2021.
Programs
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Maple
a:= proc(n) option remember; `if`(n<3, 1, add(a(n-k)*add(d* a(d)^(k/d), d=numtheory[divisors](k)), k=1..n-2)/(n-2)) end: seq(a(n), n=1..37); # Alois P. Heinz, Jul 01 2021 -
Mathematica
a[n_] := a[n] = SeriesCoefficient[x + x^2/Product[(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 37}] a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 37}]
Formula
G.f.: x + x^2 * exp(Sum_{n>=1} Sum_{k>=1} a(n)^k * x^(n*k) / k).
a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d)^(k/d) ) * a(n-k+2).