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

Author

Jon Perry, Mar 30 2004

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A027349.

Programs

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 *)

PARI

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

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