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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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.

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).

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A121833 A091606 A027037 * A020878 A158278 A187505

Adjacent sequences: A276425 A276426 A276427 * A276429 A276430 A276431

Keyword

nonn

Author

Emeric Deutsch, Sep 19 2016

Status

approved