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

0,3

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

Formula

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

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 *)

Crossrefs

Cf. A006128, A070933, A256193.

Sequence in context: A201478 A176470 A240693 * A349974 A186254 A158896

Adjacent sequences: A278461 A278462 A278463 * A278465 A278466 A278467

Keyword

nonn

Author

Alois P. Heinz, Nov 22 2016

Status

approved