A184645 Number of partitions of n having no parts with multiplicity 10.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 56, 76, 100, 133, 174, 227, 293, 378, 482, 614, 777, 980, 1229, 1538, 1913, 2375, 2936, 3619, 4445, 5447, 6650, 8102, 9844, 11929, 14421, 17397, 20934, 25141, 30130, 36035, 43014, 51253, 60952, 72367, 85771, 101488

Offset: 0

Author

Alois P. Heinz, Jan 18 2011

Keywords

• nonn

Comments

In general, if k>=1 and g.f. = Product_{j>0} (1 - x^(k*j) + x^((k+1)*j)) / (1-x^j), then 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(-k*x) + exp(-(k+2)*x)) dx. - Vaclav Kotesovec, Jun 12 2025

Links

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

Crossrefs

Cf. A000041, A183567, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=10, [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 == 10, {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 *)

Formula

a(n) = A000041(n) - A183567(n).

a(n) = A183568(n,0) - A183568(n,10).

G.f.: Product_{j>0} (1-x^(10*j)+x^(11*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(-10*x) + exp(-12*x)) dx = 0.81326581550954971049947225462608121545474493920551191360132... - Vaclav Kotesovec, Jun 12 2025