A116932 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times.

1, 2, 2, 3, 3, 6, 6, 9, 12, 14, 16, 24, 25, 32, 40, 49, 56, 73, 81, 102, 120, 142, 162, 202, 227, 270, 316, 367, 419, 506, 565, 663, 767, 879, 998, 1179, 1317, 1517, 1739, 1979, 2232, 2588, 2883, 3295, 3742, 4220, 4737, 5426, 6037, 6828, 7701, 8642, 9651, 10939

Offset: 1

Author

Emeric Deutsch, Feb 27 2006

Keywords

• nonn

Comments

Also, partitions of n in which any two distinct parts differ by at least 3. Example: a(5) = 3 because we have [5], [4,1] and [1,1,1,1,1].

Examples

			a(5) = 3 because we have [5], [2,1,1,1] and [1,1,1,1,1].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A116931, A100405.

Column k=3 of A218698. - Alois P. Heinz, Nov 04 2012

Programs

Maple

g:=sum(x^k*product(1+x^(3*j)/(1-x^j),j=1..k-1)/(1-x^k),k=1..70): gser:=series(g,x=0,62): seq(coeff(gser,x^n),n=1..58);
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-3), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 04 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-3], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

Formula

G.f.: sum(x^k*product(1+x^(3j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.

log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869... - Vaclav Kotesovec, Jan 28 2022