Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Nov 02 2023 10:30:19
%S 1,3,8,20,49,119,289,705,1731,4283,10690,26934,68531,176115,457110,
%T 1198128,3170607,8468277,22818167,61999531,169778889,468292663,
%U 1300270333,3632269293,10202425207,28798822159,81652955889,232429744843,663969970203,1902716831527
%N Binomial transform of A000029.
%H Alois P. Heinz, <a href="/A054192/b054192.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ 3^(n+1) / (4*n). - _Vaclav Kotesovec_, Nov 02 2023
%p with(numtheory):
%p b:= proc(n) option remember; ceil(add(
%p phi(d)*2^(n/d)/(2*n), d=divisors(n))+
%p `if`(n::odd, 2^((n-1)/2), 2^(n/2-1)+2^(n/2-2)))
%p end:
%p a:= n-> add(b(n-j)*binomial(n, j), j=0..n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 17 2017
%t a29[n_] := If[n == 0, 1, DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n) + If[OddQ[n], 2^((n-1)/2), 2^(n/2-1) + 2^(n/2-2)]]; a[n_] := Sum[Binomial[n, k] * a29[k], {k, 0, n}]; Array[a, 28, 0] (* _Jean-François Alcover_, Jul 17 2017 *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Apr 29 2000