|
%I #10 Aug 12 2021 05:40:49
%S 1,1,1,2,3,6,10,20,36,73,138,281,549,1136,2263,4705,9553,20015,41096,
%T 86643,179638,380701,795892,1693003,3562217,7612680,16099538,34505797,
%U 73345831,157678081,336419942,725236780,1552662599,3354979195,7205601904,15600414855,33594465666
%N G.f.: x + x^2 / Product_{n>=1} (1 - a(n)*x^n).
%H David Callan, <a href="https://arxiv.org/abs/2108.04969">A Combinatorial Interpretation for Sequence A345973 in OEIS</a>, arXiv:2108.04969 [math.CO], 2021.
%F G.f.: x + x^2 * exp(Sum_{n>=1} Sum_{k>=1} a(n)^k * x^(n*k) / k).
%F a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d)^(k/d) ) * a(n-k+2).
%p a:= proc(n) option remember; `if`(n<3, 1, add(a(n-k)*add(d*
%p a(d)^(k/d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
%p end:
%p seq(a(n), n=1..37); # _Alois P. Heinz_, Jul 01 2021
%t 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}]
%t 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}]
%Y Cf. A007562, A032307, A093637.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Jun 30 2021
|
