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A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

%I A276434 #19 Jun 12 2025 10:11:41
%S A276434 0,0,1,0,1,1,3,3,5,6,10,12,19,23,34,41,58,72,98,121,162,200,262,323,
%T A276434 415,511,650,796,1000,1222,1522,1851,2287,2771,3399,4103,5000,6015,
%U A276434 7289,8735,10530,12579,15094,17968,21468,25477,30319,35873,42531,50177,59291
%N A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.
%H A276434 Vaclav Kotesovec, <a href="/A276434/b276434.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A276434 a(n) = Sum(k*A276433(n,k), k>=0).
%F A276434 G.f.: g(x) = Sum_(i>=1) (x^(i(i+1))(1-x^i))/Product_(i>=1) (1-x^i).
%e A276434 a(6) = 3 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2], [1',1,2,2], [2',2,2], [1,1,1,3], [1,2,3], [3,3], [1',1,4], [2,4], [1,5], [6] of 6 only the marked parts satisfy the requirement.
%p A276434 g := (sum(x^(i*(i+1))*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
%p A276434 # second Maple program:
%p A276434 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A276434      `if`(i<1, 0, add((p-> p+`if`(i+1<>j, 0,
%p A276434       [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
%p A276434     end:
%p A276434 a:= n-> b(n$2)[2]:
%p A276434 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 30 2016
%t A276434 max = 60; s = Sum[x^(i*(i+1))*(1-x^i), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 08 2016 *)
%Y A276434 Cf. A276427, A276428, A276429, A276433, A277099, A277100, A277101, A277102.
%K A276434 nonn
%O A276434 0,7
%A A276434 _Emeric Deutsch_, Sep 30 2016