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A092265 Sum of smallest parts of all partitions of n into distinct parts.

%I A092265 #37 Jan 14 2025 01:49:11
%S A092265 1,2,4,5,8,10,14,16,23,26,34,40,50,58,74,83,102,120,142,164,198,226,
%T A092265 266,308,359,412,482,548,634,730,834,950,1094,1240,1416,1609,1826,
%U A092265 2068,2350,2648,2994,3382,3806,4280,4826,5408,6070,6806,7619,8522,9534,10632
%N A092265 Sum of smallest parts of all partitions of n into distinct parts.
%H A092265 Alois P. Heinz, <a href="/A092265/b092265.txt">Table of n, a(n) for n = 1..10000</a>
%F A092265 G.f.: Sum_{n >= 1} (-1 + Product_{k >= n} 1 + x^k).
%F A092265 G.f.: Sum_{n >= 1} n*x^n*Product_{k >= n+1} (1 + x^k). - _Joerg Arndt_, Jan 29 2011
%F A092265 G.f.: Sum_{k >= 1} x^(k*(k+1)/2)/(1 - x^k)/Product_{i = 1..k} (1 - x^i). - _Vladeta Jovovic_, Aug 10 2004
%F A092265 Conjecture: a(n) = A034296(n) + A237665(n+1). - _George Beck_, May 06 2017
%F A092265 a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, May 20 2018
%p A092265 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A092265      `if`(i>n, 0, b(n,i+1)+b(n-i, i+1)))
%p A092265     end:
%p A092265 a:= n-> add(j*b(n-j, j+1), j=1..n):
%p A092265 seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 03 2016
%t A092265 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]]]; a[n_] := Sum[j*b[n - j, j + 1], {j, 1, n}]; Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jan 21 2017, after _Alois P. Heinz_ *)
%Y A092265 Cf. A046746, A005895, A006128, A092269, A092319, A092316, A034296, A237665.
%Y A092265 Cf. A026832, A336902, A336903.
%K A092265 easy,nonn
%O A092265 1,2
%A A092265 _Vladeta Jovovic_, Feb 14 2004
%E A092265 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004