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

0, 1, 0, 1, 2, 3, 3, 6, 7, 12, 15, 22, 27, 40, 49, 68, 87, 116, 145, 193, 239, 311, 387, 494, 611, 776, 952, 1193, 1464, 1817, 2214, 2733, 3315, 4060, 4911, 5974, 7195, 8713, 10448, 12585, 15048, 18039, 21486, 25660, 30462, 36231, 42888, 50820, 59972, 70843, 83354

Offset: 0

Author

Emeric Deutsch, Sep 19 2016

Keywords

• nonn

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..7500 (terms 0..1000 from Alois P. Heinz)
• Philip Cuthbertson, David J. Hemmer, Brian Hopkins, and William J. Keith, Partitions with fixed points in the sequence of first-column hook lengths, arXiv:2401.06254 [math.CO], 2024.

Crossrefs

Cf. A276427, A276434, A277101; A114638, A116861.

Programs

Maple

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);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

Mathematica

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 *)

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

Formula

a(n) = Sum_{k>=0} k*A276427(n,k).

G.f.: g(x) = Sum_{i>=1} (x^{i^2}*(1-x^i))/Product_{i>=1} (1-x^i).