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A276428 Sum over all partitions of n of the number of distinct parts i of multiplicity i.

%I A276428 #26 Jun 12 2025 11:31:45
%S A276428 0,1,0,1,2,3,3,6,7,12,15,22,27,40,49,68,87,116,145,193,239,311,387,
%T A276428 494,611,776,952,1193,1464,1817,2214,2733,3315,4060,4911,5974,7195,
%U A276428 8713,10448,12585,15048,18039,21486,25660,30462,36231,42888,50820,59972,70843,83354
%N A276428 Sum over all partitions of n of the number of distinct parts i of multiplicity i.
%H A276428 Vaclav Kotesovec, <a href="/A276428/b276428.txt">Table of n, a(n) for n = 0..7500</a> (terms 0..1000 from Alois P. Heinz)
%H A276428 Philip Cuthbertson, David J. Hemmer, Brian Hopkins, and William J. Keith, <a href="https://arxiv.org/abs/2401.06254">Partitions with fixed points in the sequence of first-column hook lengths</a>, arXiv:2401.06254 [math.CO], 2024.
%F A276428 a(n) = Sum_{k>=0} k*A276427(n,k).
%F A276428 G.f.: g(x) = Sum_{i>=1} (x^{i^2}*(1-x^i))/Product_{i>=1} (1-x^i).
%e A276428 a(5) = 3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1',2',2], [1,1,3], [2,3], [1',4], [5] of 5 only the marked parts satisfy the requirement.
%p A276428 g := (sum(x^(i^2)*(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 A276428 # second Maple program:
%p A276428 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A276428      `if`(i<1, 0, add((p-> p+`if`(i<>j, 0,
%p A276428       [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
%p A276428     end:
%p A276428 a:= n-> b(n$2)[2]:
%p A276428 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 19 2016
%t A276428 b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==j, x, 1]*b[n - i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; a[n_] := (row = T[n]; row.Range[0, Length[row]-1]); Table[a[n], {n, 0, 60}] // Flatten (* _Jean-François Alcover_, Nov 28 2016, after _Alois P. Heinz_'s Maple code for A276427 *)
%o A276428 (PARI) apply( A276428(n,s,c)={forpart(p=n,c=1;for(i=1,#p,p[i]==if(i<#p, p[i+1])&&c++&&next; c==p[i]&&s++; c=1));s}, [0..20]) \\ _M. F. Hasler_, Oct 27 2019
%Y A276428 Cf. A276427, A276434, A277101; A114638, A116861.
%K A276428 nonn
%O A276428 0,5
%A A276428 _Emeric Deutsch_, Sep 19 2016