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Binomial transform of {A004111(n), n >= 1}.
5

%I #17 Oct 30 2017 05:10:59

%S 1,2,4,9,22,57,155,439,1283,3837,11675,36013,112348,353836,1123431,

%T 3591616,11551046,37343096,121280307,395496997,1294457887,4250811199,

%U 14001176036,46243806379,153123238870,508207709138,1690355937970,5633580018286,18810483711103,62917378114528,210788885780702,707273100413094

%N Binomial transform of {A004111(n), n >= 1}.

%H Alois P. Heinz, <a href="/A196161/b196161.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 1 + A246169 = 3.51754035263200389079535459..., c = 0.428531715886712592684516703... - _Vaclav Kotesovec_, Oct 30 2017

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(

%p b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))

%p end:

%p a:= n-> add(b(k+1)*binomial(n-1, k), k=0..n-1):

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Feb 24 2015

%t 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_ *)

%Y Cf. A004111, A196154.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Oct 27 2011