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A309371 a(n) = Sum_{k=1..n} k * A088370(n,k).

%I A309371 #15 Sep 15 2022 04:08:39
%S A309371 0,1,5,13,29,48,82,122,186,239,327,419,559,674,852,1028,1284,1453,
%T A309371 1721,1977,2353,2636,3062,3462,4030,4403,4971,5495,6243,6790,7592,
%U A309371 8328,9352,9945,10861,11685,12869,13704,14938,16050,17602,18567,20015,21307,23127,24410
%N A309371 a(n) = Sum_{k=1..n} k * A088370(n,k).
%H A309371 Alois P. Heinz, <a href="/A309371/b309371.txt">Table of n, a(n) for n = 0..10000</a>
%F A309371 a(n) = Sum_{k=1..n} k * A088370(n,k).
%F A309371 A000292(n) <= a(n) <= A000330(n).
%p A309371 b:= proc(n) option remember; `if`(n<2, n, (h->
%p A309371       [map(x-> 2*x-1, [b(n-h)])[],
%p A309371        map(x-> 2*x, [b(h)])[]][])(iquo(n, 2)))
%p A309371     end:
%p A309371 a:= n-> (l-> add(i*l[i], i=1..n))([b(n)]):
%p A309371 seq(a(n), n=0..50);
%t A309371 T[n_] := T[n] = If[n == 1, {1}, Join[q = Quotient[n, 2];
%t A309371    2*T[n - q] - 1, 2*T[q]]];
%t A309371 a[n_] := Sum[k*T[n][[k]], {k, 1, n}];
%t A309371 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Sep 15 2022, after _Alois P. Heinz_ in A088370 *)
%Y A309371 Cf. A000292, A000330, A088370.
%K A309371 nonn
%O A309371 0,3
%A A309371 _Alois P. Heinz_, Jul 25 2019