A327607 Number of parts in all twice partitions of n where the first partition is strict.

0, 1, 3, 11, 21, 58, 128, 276, 516, 1169, 2227, 4324, 8335, 15574, 29116, 55048, 97698, 176291, 323277, 563453, 1005089, 1770789, 3076868, 5293907, 9184885, 15668638, 26751095, 45517048, 76882920, 128738414, 217219751, 360525590, 599158211, 995474365

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Example

a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.

Maple

g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
      numtheory[tau](j), j=1..n)])(combinat[numbpart])
    end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
      `if`(n=0, [1, 0], b(n, i-1) +(h-> (f-> f+[0, f[1]*
       h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);

Mathematica

g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000009, A000041, A271619, A327552, A327594, A327605, A327608.

Sequence in context: A064568 A147073 A147191 * A034185 A184053 A240371

Adjacent sequences: A327604 A327605 A327606 * A327608 A327609 A327610

Keyword

nonn

Author

Alois P. Heinz, Sep 18 2019

Status

approved