OFFSET
0,4
COMMENTS
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - Seiichi Manyama, Nov 14 2018
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
FORMULA
G.f.: Product_{k>0} (1 + A000009(k)*x^k). - Seiichi Manyama, Nov 14 2018
EXAMPLE
The a(6)=11 twice-partitions are:
((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).
MAPLE
with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..70); # Alois P. Heinz, Dec 20 2016
MATHEMATICA
nn=20; CoefficientList[Series[Product[(1+PartitionsQ[k]x^k), {k, nn}], {x, 0, nn}], x]
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n]; b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 18 2016
STATUS
approved