A116645 -id:A116645 - cp's OEIS frontend

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

Alois P. Heinz, Jan 18 2011

nonn

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

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

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

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);

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 *)

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

A116644 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k doubletons (n>=0, k>=0). By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 2, 5, 2, 8, 2, 1, 10, 5, 13, 8, 1, 20, 9, 1, 26, 12, 4, 33, 21, 2, 46, 25, 5, 1, 58, 37, 6, 75, 48, 11, 1, 101, 59, 16, 125, 84, 19, 3, 157, 115, 23, 2, 206, 135, 39, 5, 253, 187, 46, 4, 317, 238, 63, 8, 1, 403, 292, 90, 7, 494, 382, 108, 17, 1, 608, 490, 139, 18

Offset: 0

Emeric Deutsch, Feb 20 2006

Apparently, rows n with p(p+1)<=n<(p+1)(p+2) have at most p+1 terms. Row sums are the partition numbers (A000041). T(n,0)=A116645(n). Sum(k*T(n,k),k>=0)=A116646(n).

			T(6,2) = 1 because [2,2,1,1] is the only partition of 6 with 2 doubletons.
Triangle starts:
1;
1;
1,  1;
3;
3,  2;
5,  2;
8,  2, 1;
10, 5;
13, 8, 1;

Alois P. Heinz, Rows n = 0..614, flattened

Cf. A000041, A116645, A116646.

g:=product(1+x^j+t*x^(2*j)+x^(3*j)/(1-x^j),j=1..35): gser:=simplify(series(g,x=0,35)): P[0]:=1: for n from 1 to 24 do P[n]:=coeff(gser,x^n) od: for n from 0 to 24 do seq(coeff(P[n],t,j),j=0..degree(P[n])) od; # sequence given in triangular form

G.f.: G(t,x) = product(1+x^j+tx^(2j)+x^(3j)/(1-x^j), j=1..infinity).

cp's OEIS Frontend

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

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