OFFSET
2,2
COMMENTS
With offset 0 convolution square of A000009(k+1). - George Beck, Jan 28 2021
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 2..10000 (terms 2..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019
G.f.: (-1 + Product_{j>=1} (1 + x^j))^2. - Alois P. Heinz, Jan 29 2021
EXAMPLE
a(4) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
a(5) = 8: 2a2b1a, 2a2b1b, 3a1a1b, 3b1a1b, 3a2b, 3b2a, 4a1b, 4b1a.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..45);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 45] (* Jean-François Alcover, May 06 2020, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 03 2019
STATUS
approved