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

%I A278464 #18 Nov 10 2017 08:00:30
%S A278464 0,1,5,17,53,145,385,957,2333,5493,12741,28941,65049,144225,317229,
%T A278464 691457,1497901,3224145,6906969,14726701,31282421,66211253,139720445,
%U A278464 294007373,617154865,1292516577,2701451621,5635565761,11736442005,24403092657,50666528209
%N A278464 Total number of parts of the second sort in all partitions of n into two sorts of parts.
%C A278464 a(n) is odd for n > 0.
%H A278464 Alois P. Heinz, <a href="/A278464/b278464.txt">Table of n, a(n) for n = 0..3309</a>
%H A278464 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 A278464 a(n) = Sum_{k=0..n} k * A256193(n,k).
%p A278464 b:= proc(n, i) option remember; `if`(n=0, [1/2, 0], `if`(i<1, 0,
%p A278464       b(n, i-1) +`if`(i>n, 0, (p-> p+[0, p[1]])(2*b(n-i, i)))))
%p A278464     end:
%p A278464 a:= n-> b(n$2)[2]:
%p A278464 seq(a(n), n=0..35);
%t A278464 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 A278464 a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]] . Range[0, n];
%t A278464 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 10 2017, after _Alois P. Heinz_ *)
%Y A278464 Cf. A006128, A070933, A256193.
%K A278464 nonn
%O A278464 0,3
%A A278464 _Alois P. Heinz_, Nov 22 2016