A278464 Total number of parts of the second sort in all partitions of n into two sorts of parts.

0, 1, 5, 17, 53, 145, 385, 957, 2333, 5493, 12741, 28941, 65049, 144225, 317229, 691457, 1497901, 3224145, 6906969, 14726701, 31282421, 66211253, 139720445, 294007373, 617154865, 1292516577, 2701451621, 5635565761, 11736442005, 24403092657, 50666528209

Offset: 0

Author

Alois P. Heinz, Nov 22 2016

Keywords

• nonn

Comments

a(n) is odd for n > 0.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3309
• William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017

Crossrefs

Cf. A006128, A070933, A256193.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1/2, 0], `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, (p-> p+[0, p[1]])(2*b(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..35);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]] . Range[0, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A256193(n,k).