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A279785
Number of ways to choose a strict partition of each part of a strict partition of n.
47
1, 1, 1, 3, 4, 7, 11, 18, 28, 47, 71, 108, 166, 252, 382, 587, 869, 1282, 1938, 2832, 4153, 6148, 8962, 12965, 18913, 27301, 39380, 56747, 81226, 115907, 166358, 236000, 334647, 475517, 671806, 947552, 1335679, 1875175, 2630584, 3687589, 5150585, 7183548
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