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a(n) = Sum_{k>0} k * A326616(k,n).
3

%I #28 Mar 24 2022 07:47:28

%S 0,1,14,243,9692,865445,196868202,122831606807,219073289264824,

%T 1139077903664789577,17597009238919048388550,

%U 821444189426979675481201211,116802449602563244067365434335892,50816512870344533477388136382624158445

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

%H Alois P. Heinz, <a href="/A326651/b326651.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = Sum_{k=n..A024916(n)} k * A326616(k,n) = Sum_{k=n..A024916(n)} k * A326617(k,n).

%p g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->

%p b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))

%p end:

%p a:= k-> add(n*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):

%p seq(a(n), n=0..15);

%t g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];

%t 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 a[k_] := Sum[n*Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}], {n, k, g[k]}];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 24 2022, after _Alois P. Heinz_ *)

%Y Cf. A000203, A024916, A326616, A326617.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 12 2019