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A326651 a(n) = Sum_{k>0} k * A326616(k,n).

%I A326651 #28 Mar 24 2022 07:47:28
%S A326651 0,1,14,243,9692,865445,196868202,122831606807,219073289264824,
%T A326651 1139077903664789577,17597009238919048388550,
%U A326651 821444189426979675481201211,116802449602563244067365434335892,50816512870344533477388136382624158445
%N A326651 a(n) = Sum_{k>0} k * A326616(k,n).
%H A326651 Alois P. Heinz, <a href="/A326651/b326651.txt">Table of n, a(n) for n = 0..50</a>
%F A326651 a(n) = Sum_{k=n..A024916(n)} k * A326616(k,n) = Sum_{k=n..A024916(n)} k * A326617(k,n).
%p A326651 g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
%p A326651 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A326651       b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
%p A326651     end:
%p A326651 a:= k-> add(n*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):
%p A326651 seq(a(n), n=0..15);
%t A326651 g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
%t A326651 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i*j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
%t A326651 a[k_] := Sum[n*Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}], {n, k, g[k]}];
%t A326651 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 24 2022, after _Alois P. Heinz_ *)
%Y A326651 Cf. A000203, A024916, A326616, A326617.
%K A326651 nonn
%O A326651 0,3
%A A326651 _Alois P. Heinz_, Sep 12 2019