A092130 Number of partitions of n into distinct parts == 1 (mod 3), with 1 as the smallest part.

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 4, 3, 1, 4, 4, 2, 5, 5, 2, 5, 7, 3, 6, 8, 4, 6, 10, 6, 7, 12, 7, 8, 14, 10, 9, 16, 12, 10, 19, 16, 12, 21, 19, 14, 24, 24, 17, 27, 28, 20, 31, 35, 24, 34, 40, 29, 39, 48, 35

Offset

1,18

Comments

Also number of partitions of n such that if k is the largest part, then k occurs exactly once and integers from 1 to k-1 occur a positive multiple of 3 times. Example: a(18)=2 because we have [3,2,2,2,1,1,1,1,1,1,1,1,1] and [3,2,2,2,2,2,2,1,1,1]. - Emeric Deutsch, Apr 18 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: x*Product_{k>=1} (1+x^(1+3k)). - Emeric Deutsch, Apr 18 2006

a(n) ~ exp(Pi*sqrt(n)/3) / (2^(7/3) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

G.f.: Sum_{k>=1} x^(k*(3*k - 1)/2) / Product_{j=1..k-1} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 28 2020

Example

For a(24), we have 19+4+1, 16+7+1, 13+10+1, so a(24)=3.

Maple

g:=x*product(1+x^(1+3*k), k=1..25): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=1..51); # Emeric Deutsch, Apr 18 2006
# second Maple program
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-3)+`if`(i>n, 0, b(n-i, i-3))))
    end:
a:= n-> b(n-1, iquo(n, 3)*3+1):
seq (a(n), n=1..100);  # Alois P. Heinz, May 01 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-3] + If[i>n, 0, b[n-i, i-3]]]]; a[n_] := b[n-1, Quotient[n, 3]*3+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)

Prog

(PARI) for(i=0, 50, print1(", "polcoeff(prod(k=1, 50, (1+x^(3*k+1))), i)))

Crossrefs

Cf. A027349.

Sequence in context: A025847 A367771 A334152 * A029298 A262520 A351074

Adjacent sequences: A092127 A092128 A092129 * A092131 A092132 A092133

Keyword

nonn

Author

Jon Perry, Mar 30 2004

Status

approved