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A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.

%I A277101 #14 Jun 12 2025 10:53:02
%S A277101 0,0,1,1,1,2,4,5,8,10,15,20,29,37,52,67,89,115,152,192,251,316,405,
%T A277101 508,644,799,1006,1243,1546,1901,2351,2871,3527,4289,5232,6336,7688,
%U A277101 9264,11189,13430,16137,19299,23097,27514,32799,38944,46246,54738,64782,76430,90171
%N A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.
%H A277101 Vaclav Kotesovec, <a href="/A277101/b277101.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A277101 a(n) = Sum(k*A277100(n,k), k>=0).
%F A277101 G.f.: g(x) = Sum_(i>=1)(x^(i(i+1))(1-x^(i+1)))/Product_(i>=1)(1-x^i).
%e A277101 a(6) = 4 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 A277101 g := (sum(x^(i*(i+1))*(1-x^(i+1)), 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 A277101 # second Maple program:
%p A277101 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A277101      `if`(i<1, 0, add((p-> p+`if`(i-1<>j, 0,
%p A277101       [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
%p A277101     end:
%p A277101 a:= n-> b(n$2)[2]:
%p A277101 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 10 2016
%t A277101 max = 60; s = Sum[x^(i*(i+1))*(1-x^(1+i)), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 08 2016 *)
%Y A277101 Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277102.
%K A277101 nonn
%O A277101 0,6
%A A277101 _Emeric Deutsch_, Oct 10 2016