A196161 Binomial transform of {A004111(n), n >= 1}.
1, 2, 4, 9, 22, 57, 155, 439, 1283, 3837, 11675, 36013, 112348, 353836, 1123431, 3591616, 11551046, 37343096, 121280307, 395496997, 1294457887, 4250811199, 14001176036, 46243806379, 153123238870, 508207709138, 1690355937970, 5633580018286, 18810483711103, 62917378114528, 210788885780702, 707273100413094
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
Programs
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Maple
with(numtheory): b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add( b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1)) end: a:= n-> add(b(k+1)*binomial(n-1, k), k=0..n-1): seq(a(n), n=1..50); # Alois P. Heinz, Feb 24 2015 -
Mathematica
b[n_] := b[n] = If[n < 2, n, Sum[b[n - k]*Sum[ b[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n-1}]/(n-1)]; a[n_] := Sum[b[k+1]*Binomial[n-1, k], {k, 0, n-1}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 12 2016, after Alois P. Heinz *)
Formula
a(n) ~ c * d^n / n^(3/2), where d = 1 + A246169 = 3.51754035263200389079535459..., c = 0.428531715886712592684516703... - Vaclav Kotesovec, Oct 30 2017