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A085811 Number of partitions of n including 3, but not 1. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 17 08:39 EST 2019. Contains 329217 sequences. (Running on oeis4.)