%I #25 Jan 20 2023 16:36:53
%S 0,1,5,13,32,61,123,208,367,590,957,1459,2266,3328,4938,7097,10205,
%T 14299,20100,27626,38023,51485,69600,92882,123863,163235,214798,
%U 280141,364530,470660,606557,776233,991370,1258827,1594741,2010142,2528445,3165648,3955190
%N Sum of the first moments of all partitions of n.
%C The first element of each partition is given weight 1.
%H Alois P. Heinz, <a href="/A066184/b066184.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = 1/2*(A066183(n) + A066186(n)). - _Vladeta Jovovic_, Mar 23 2003
%F G.f.: Sum_{k>=1} x^k/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%e a(3)=13 because the first moments of all partitions of 3 are {3}.{1},{2,1}.{1,2} and {1,1,1}.{1,2,3}, resulting in 3,4,6; summing to 13.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p b(n, i-1)+(h-> h+[0, h[1]*i*(i+1)/2])(b(n-i, min(n-i, i))))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 29 2014
%t Table[ Plus@@ Map[ #.Range[ Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
%t (* Second program: *)
%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, If[i > n, b[n, i - 1], b[n, i - 1] + Function[h, h + {0, h[[1]]*i*(i + 1)/2}][b[n - i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y Cf. A066185.
%K easy,nonn
%O 0,3
%A _Wouter Meeussen_, Dec 15 2001