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

0,4

Links

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

Example

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

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<n, 0,
     `if`(n=0, [1, 0], b(n, i-1)+(p-> 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 *)

Crossrefs

Cf. A000009, A279785, A327605.

Sequence in context: A047811 A244010 A154145 * A302428 A336142 A091280

Adjacent sequences: A327550 A327551 A327552 * A327554 A327555 A327556

Keyword

nonn

Author

Alois P. Heinz, Sep 16 2019

Status

approved