A184645 - OEIS

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

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OFFSET

0,3

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

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

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 *)

CROSSREFS