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A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n).

%I A057623 #36 Nov 07 2020 10:40:41
%S A057623 1,5,29,218,1814,18144,196356,2427312,32304240,475637760,7460546400,
%T A057623 127525829760,2302819079040,44659367020800,911770840108800,
%U A057623 19784985947596800,449672462639769600,10790180876185804800,270071861749240320000,7094011359005190144000
%N A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n).
%H A057623 Alois P. Heinz, <a href="/A057623/b057623.txt">Table of n, a(n) for n = 1..400</a>
%H A057623 Guo-Niu Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO]; see p.27
%F A057623 n! *sum_{k=1 to n} [sigma(k) p(n-k) /k], where sigma(n) = sum of positive divisors of n and p(n) = number of unrestricted partitions of n.
%F A057623 a(n) = P(n,1), where P(n,m) = P(n,m+1)+S(n-m,m)*n!/m+n!/(n-m)!*P(n-m,m)), P(n,n)=(n-1)!, P(n,m)=0 for m>n, S(n,m) is triangle of A026807. - _Vladimir Kruchinin_, Sep 10 2014
%e A057623 The unrestricted partitions of 3 are 1 + 1 + 1, 1 + 2 and 3. So a(3) = 3! *(1 + 1 + 1 + 1 + 1/2 + 1/3) = 29.
%p A057623 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A057623        b(n, i-1)+`if`(i>n, 0, (p-> p+[0, p[1]/i])(b(n-i, i)))))
%p A057623     end:
%p A057623 a:= n-> n!*b(n$2)[2]:
%p A057623 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 11 2014
%t A057623 b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[ {p}, p + {0, p[[1]]/i}][b[n-i, i]]]]]; a[n_] := n!*b[n, n][[2]]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Apr 02 2015, after _Alois P. Heinz_ *)
%t A057623 Table[n!*Sum[DivisorSigma[1, k]*PartitionsP[n - k]/k, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, May 29 2018 *)
%o A057623 (Maxima)
%o A057623 S(n,m):=(if n=m then 1 else if n<m then 0 else S(n,m+1)+S(n-m,m));
%o A057623 P(n,m):=(if n=m then (n-1)! else if n<m then 0 else P(n,m+1)+S(n-m,m)*n!/(m)+(n!)/(n-m)!*P(n-m,m));
%o A057623 makelist(P(n,1),n,1,18); /* _Vladimir Kruchinin_, Sep 10 2014 */
%o A057623 (PARI) {a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-1-t)), n), 1)} \\ _Seiichi Manyama_, Nov 07 2020
%Y A057623 Column 1 of A210590.
%Y A057623 Cf. A103738.
%K A057623 nonn
%O A057623 1,2
%A A057623 _Leroy Quet_, Oct 09 2000