A196154 Binomial transform of A004111.
0, 1, 3, 7, 16, 38, 95, 250, 689, 1972, 5809, 17484, 53497, 165845, 519681, 1643112, 5234728, 16785774, 54128870, 175409177, 570906174, 1865364061, 6116175260, 20117351296, 66361157675, 219484396545, 727692105683, 2418048043653, 8051628061939, 26862111773042, 89779489887570, 300568375668272, 1007841476081366
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..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)*binomial(n, k), k=0..n): seq(a(n), n=0..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]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 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 = 0.59875012586719098912050580024... - Vaclav Kotesovec, Oct 30 2017