A184642 Number of partitions of n having no parts with multiplicity 7.

1, 1, 2, 3, 5, 7, 11, 14, 22, 29, 41, 54, 75, 97, 130, 168, 222, 283, 368, 465, 597, 750, 949, 1183, 1488, 1841, 2292, 2822, 3487, 4267, 5239, 6376, 7782, 9429, 11439, 13798, 16661, 20007, 24043, 28763, 34420, 41021, 48894, 58066, 68956, 81627, 96592

Offset: 0

Author

Alois P. Heinz, Jan 18 2011

Keywords

• nonn

Examples

			a(7) = 14, because 14 partitions of 7 have no parts with multiplicity 7: [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,1,3], [1,1,2,3], [2,2,3], [1,3,3], [1,1,1,4], [1,2,4], [3,4], [1,1,5], [2,5], [1,6], [7].
		

Links

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

Crossrefs

Cf. A000041, A183564, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184643, A184644, A184645.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=7, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 7, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(FreeQ[Length/@Split[#],7]&)],{n,0,50}] (* _Harvey P. Dale, Sep 21 2024 *)

Formula

a(n) = A000041(n) - A183564(n).

a(n) = A183568(n,0) - A183568(n,7).

G.f.: Product_{j>0} (1-x^(7*j)+x^(8*j))/(1-x^j).

a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-7*x) + exp(-9*x)) dx = 0.80430417180776436899064351977235191494130305607975798117531... - Vaclav Kotesovec, Jun 12 2025