This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066184 #30 Jul 06 2025 08:41:13
%S A066184 0,1,5,13,32,61,123,208,367,590,957,1459,2266,3328,4938,7097,10205,
%T A066184 14299,20100,27626,38023,51485,69600,92882,123863,163235,214798,
%U A066184 280141,364530,470660,606557,776233,991370,1258827,1594741,2010142,2528445,3165648,3955190
%N A066184 Sum of the first moments of all partitions of n with weight starting at 1.
%C A066184 The first element of each partition is given weight 1.
%H A066184 Alois P. Heinz, <a href="/A066184/b066184.txt">Table of n, a(n) for n = 0..10000</a>
%F A066184 a(n) = 1/2*(A066183(n) + A066186(n)). - _Vladeta Jovovic_, Mar 23 2003
%F A066184 G.f.: Sum_{k>=1} x^k/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%F A066184 a(n) ~ 3 * zeta(3) * sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2) * Pi^3). - _Vaclav Kotesovec_, Jul 06 2025
%e A066184 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 A066184 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p A066184 b(n, i-1)+(h-> h+[0, h[1]*i*(i+1)/2])(b(n-i, min(n-i, i))))
%p A066184 end:
%p A066184 a:= n-> b(n$2)[2]:
%p A066184 seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 29 2014
%t A066184 Table[ Plus@@ Map[ #.Range[ Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
%t A066184 (* Second program: *)
%t A066184 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 A066184 Cf. A066185.
%K A066184 easy,nonn
%O A066184 0,3
%A A066184 _Wouter Meeussen_, Dec 15 2001