A085811 Number of partitions of n including 3, but not 1.

0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701

Offset

1,7

Comments

Related to the 'number of sums containing k' phenomena reported at link. Define P_k(n,j) to be the number of partitions of n with minimum part j and containing k, P_k(n) as the number of partitions of n that contain k as a part and P(n,j) as the number of partitions of k that have minimum part k, then: P_k(n)=sum{i=1,k-1,P_k(n-i,i)}+P(n-k,k) which (unproved) gives P(n-k). This sequence gives P_3(n,2). E.g. assume P_3(9)=11. P_3(10)=P_3(9,1)+P_3(8,2)+P(7,3)=11+2+2=15, where P(7,3) is given by A008483(7).

Links

Andrew van den Hoeven, Table of n, a(n) for n = 1..10000

Formula

A002865(n) = a(n+3). - James A. Sellers, Dec 06 2005.

G.f.: x^3*(1-x)/prod(n>=1, 1-x^n). [Joerg Arndt, Feb 03 2012]

G.f.: x^2 + x^3*(1 - G(0))/(1-x) where G(k) = 1 - x/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013

Example

a(3): 3
a(5): 2+3
a(6): 3+3
a(7): 2+2+3, 3+4
a(8): 2+3+3, 3+5
a(9): 2+3+4, 2+2+2+3, 3+3+3, 3+6
a(10): 2+3+5, 2+2+3+3, 3+7, 3+3+4
a(11): 2+2+3+4, 2+3+6, 2+2+2+2+3, 2+3+3+3, 3+4+4, 3+8, 3+3+5,
a(12): 2+2+2+3+3, 2+3+3+4, 2+3+7, 2+2+3+5, 3+9, 3+3+6, 3+4+5, 3+3+3+3
a(13): 2+2+2+2+2+3, 2+2+2+3+4, 2+2+3+6, 2+2+3+3+3, 2+3+4+4, 2+3+3+5,
2+3+8, 3+10, 3+3+7, 3+4+6, 3+5+5, 3+3+3+4

Mathematica

f[n_] := Block[{c = 0, k = 1, m = PartitionsP[n], p = IntegerPartitions[n] }, While[k < m, If[ Count[ p[[k]], 3] > 0 && Count[ p[[k]], 1] == 0, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 53}]
(* second program: *)
CoefficientList[x^2*(1-x)/QPochhammer[x] + O[x]^60, x] (* Jean-François Alcover, Jan 22 2016, after Joerg Arndt *)

Prog

(PARI)  x='x+O('x^66); /* about that many terms */
v=Vec((x^3*(1-x)/eta(x)))  /* Joerg Arndt, Feb 03 2012 */

Crossrefs

Cf. A008483, A002865. Essentially the same as A002865.

Sequence in context: A240019 A036000 A002865 * A187219 A317785 A014810

Adjacent sequences: A085808 A085809 A085810 * A085812 A085813 A085814

Keyword

nonn

Author

Jon Perry, Jul 25 2003

Extensions

Edited, corrected and extended by Robert G. Wilson v

Typo in formula corrected by Andrew van den Hoeven, Nov 20 2014

Status

approved