A083751 Number of partitions of n into >= 2 parts and with minimum part >= 2.

0, 0, 0, 1, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset: 1

Author

Jon Perry, Jun 17 2003

Keywords

• nonn

Comments

Also number of partitions of n such that the largest part is at least 2 and occurs at least twice. Example: a(6)=3 because we have [3,3],[2,2,2] and [2,2,1,1]. - Emeric Deutsch, Mar 29 2006

Also number of partitions of n that contain emergent parts (Cf. A182699). - Omar E. Pol, Oct 21 2011

Also number of regions in the last section of the set of partitions of n that do not contain 1 as a part (cf. A187219). - Omar E. Pol, Mar 04 2012

Schneider calls these "nuclear partitions" and gives a remarkable formula relating a(n), the number of partitions of n, and a sum over the two greatest parts of each such partition. - Charles R Greathouse IV, Dec 04 2019

Examples

			a(6) = 3, as 6 = 2+4 = 3+3 = 2+2+2.
a(6) = 3 because 6 = 2+4 = 3+3 = 2+2+2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Robert Schneider, Nuclear partitions and a formula for p(n), arXiv:1912.00575 [math.NT], 2019.

Crossrefs

Cf. A053445, A072380, A008483, A026796, A035989, A036000, A002865, A081094.

First differences of A000094.

Programs

Maple

g:=sum(x^(2*j)/product(1-x^i,i=1..j),j=2..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Mar 29 2006

Mathematica

Drop[CoefficientList[Series[1/Product[(1-x^k)^1, {k, 2, 50}], {x, 0, 50}], x]-1, 2]
(* or *) Table[Count[IntegerPartitions[n], q_List /; Length[q] > 1 && Min[q] >= 2 ], {n, 24}]

Formula

a(n) = A000041(n) - A000041(n-1) - 1, n > 1. - Vladeta Jovovic, Jun 18 2003

G.f.: Sum_{j>=2} x^(2j)/Product_{i=1..j} (1-x^i). - Emeric Deutsch, Mar 29 2006

a(n) = A002865(n) - 1, n > 1. - Omar E. Pol, Oct 21 2011

a(n) = A187219(n) - 1. - Omar E. Pol, Mar 04 2012

Extensions

More terms from Vladeta Jovovic and Wouter Meeussen, Jun 18 2003

Description corrected by James Sellers, Jun 21 2003