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A196154 Binomial transform of A004111.

%I A196154 #22 Oct 30 2017 05:07:03
%S A196154 0,1,3,7,16,38,95,250,689,1972,5809,17484,53497,165845,519681,1643112,
%T A196154 5234728,16785774,54128870,175409177,570906174,1865364061,6116175260,
%U A196154 20117351296,66361157675,219484396545,727692105683,2418048043653,8051628061939,26862111773042,89779489887570,300568375668272,1007841476081366
%N A196154 Binomial transform of A004111.
%H A196154 Alois P. Heinz, <a href="/A196154/b196154.txt">Table of n, a(n) for n = 0..1000</a>
%F A196154 a(n) ~ c * d^n / n^(3/2), where d = 1 + A246169 = 3.51754035263200389079535459..., c = 0.59875012586719098912050580024... - _Vaclav Kotesovec_, Oct 30 2017
%p A196154 with(numtheory):
%p A196154 b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(
%p A196154       b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
%p A196154     end:
%p A196154 a:= n-> add(b(k)*binomial(n, k), k=0..n):
%p A196154 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 24 2015
%t A196154 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]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 12 2016, after _Alois P. Heinz_ *)
%Y A196154 Cf. A004111, A196161.
%K A196154 nonn
%O A196154 0,3
%A A196154 _N. J. A. Sloane_, Oct 27 2011