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A066185 Sum of the first moments of all partitions of n with weights starting at 0.

%I A066185 #50 Jul 06 2025 08:41:21
%S A066185 0,0,1,4,12,26,57,103,191,320,537,843,1342,2015,3048,4457,6509,9250,
%T A066185 13170,18316,25483,34853,47556,64017,86063,114285,151462,198871,
%U A066185 260426,338275,438437,564131,724202,924108,1176201,1489237,1881273,2365079,2966620,3705799
%N A066185 Sum of the first moments of all partitions of n with weights starting at 0.
%C A066185 The first element of each partition is given weight 0.
%C A066185 Consider the partitions of n, e.g., n=5. For each partition sum T(e-1) and sum all these. E.g., 5 -> T(4)=10, 41 -> T(3)+T(0)=6, 32 -> T(2)+T(1)=4, 311 -> T(2)+T(0)+T(0)=3, 221 -> T(1)+T(1)+T(0)=2, 21111 ->1 and 11111 ->0. Summing, 10+6+4+3+2+1+0 = 26 as desired. - _Jon Perry_, Dec 12 2003
%C A066185 Also equals the sum of f(p) over the partitions p of n, where f(p) is obtained by replacing each part p_i of partition p by p_i*(p_i-1)/2. See I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition, p. 3, eqn (1.5) and (1.6). - _Wouter Meeussen_, Sep 25 2014
%H A066185 Alois P. Heinz, <a href="/A066185/b066185.txt">Table of n, a(n) for n = 0..10000</a>
%F A066185 a(n) = 1/2*(A066183(n) - A066186(n)). - _Vladeta Jovovic_, Mar 23 2003
%F A066185 G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Mar 05 2021
%F A066185 a(n) = Sum_{k=0..A161680(n)} k * A264034(n,k). - _Alois P. Heinz_, Jan 20 2023
%F A066185 a(n) ~ 3 * zeta(3) * sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2) * Pi^3). - _Vaclav Kotesovec_, Jul 06 2025
%e A066185 a(3)=4 because the first moments of all partitions of 3 are {3}.{0},{2,1}.{0,1} and {1,1,1}.{0,1,2}, resulting in 0,1,3; summing to 4.
%p A066185 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0],
%p A066185       b(n, i-1)+(h-> h+[0, h[1]*i*(i-1)/2])(b(n-i, min(n-i, i))))
%p A066185     end:
%p A066185 a:= n-> b(n$2)[2]:
%p A066185 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 29 2014
%t A066185 Table[ Plus@@ Map[ #.Range[ 0, -1+Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
%t A066185 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_, Oct 26 2015, after _Alois P. Heinz_ *)
%Y A066185 Cf. A000337, A001788, A066183, A066184, A066186, A161680, A264034.
%K A066185 easy,nonn
%O A066185 0,4
%A A066185 _Wouter Meeussen_, Dec 15 2001