A327553 Number of partitions in all twice partitions of n where both partitions are strict.

0, 1, 1, 4, 6, 11, 20, 33, 57, 100, 165, 254, 417, 649, 1039, 1648, 2540, 3836, 6020, 9035, 13645, 20752, 31054, 45993, 68668, 101511, 149525, 220132, 321614, 468031, 684124, 989703, 1427054, 2064859, 2964987, 4254028, 6090453, 8686574, 12366583, 17598885

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn

Examples

			a(3) = 4 = 1+1+2 counting the partitions in 3, 21, 2|1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000009, A279785, A327605.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 p+[0, p[1]])(
       g(i)*b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);

Mathematica

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, {0, 0}, If[n==0, {1, 0}, b[n, i-1] + Function[p, p + {0, p[[1]]}][g[i] b[n-i, Min[n-i, i-1]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)