A278464 - OEIS

%I #18 Nov 10 2017 08:00:30

%S 0,1,5,17,53,145,385,957,2333,5493,12741,28941,65049,144225,317229,

%T 691457,1497901,3224145,6906969,14726701,31282421,66211253,139720445,

%U 294007373,617154865,1292516577,2701451621,5635565761,11736442005,24403092657,50666528209

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

%C a(n) is odd for n > 0.

%H Alois P. Heinz, <a href="/A278464/b278464.txt">Table of n, a(n) for n = 0..3309</a>

%H William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, <a href="https://arxiv.org/abs/1702.02446">Tiered trees, weights, and q-Eulerian numbers</a>, arXiv:1702.02446 [math.CO], Feb 2017

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

%p b:= proc(n, i) option remember; `if`(n=0, [1/2, 0], `if`(i<1, 0,

%p       b(n, i-1) +`if`(i>n, 0, (p-> p+[0, p[1]])(2*b(n-i, i)))))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..35);

%t 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}]]]];

%t a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]] . Range[0, n];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 10 2017, after _Alois P. Heinz_ *)

%Y Cf. A006128, A070933, A256193.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 22 2016