A184641 Number of partitions of n having no parts with multiplicity 6.
1, 1, 2, 3, 5, 7, 10, 15, 21, 29, 40, 54, 72, 96, 127, 166, 216, 279, 358, 457, 580, 735, 924, 1159, 1446, 1799, 2228, 2752, 3388, 4158, 5087, 6207, 7551, 9165, 11093, 13401, 16144, 19412, 23286, 27882, 33310, 39727, 47289, 56191, 66647, 78923, 93299
Offset: 0
Keywords
Examples
a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0], add((l->`if`(j=6, [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 == 6, {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
G.f.: Product_{j>0} (1-x^(6*j)+x^(7*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(-6*x) + exp(-8*x)) dx = 0.79818518024793359047735154473665146019665210453617381247423... - Vaclav Kotesovec, Jun 12 2025
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