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

1,6

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

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.

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

Example

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

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}]

Crossrefs

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

First differences of A000094.

Sequence in context: A211541 A026926 A332557 * A379720 A034401 A345729

Adjacent sequences: A083748 A083749 A083750 * A083752 A083753 A083754

Keyword

nonn

Author

Jon Perry, Jun 17 2003

Extensions

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

Description corrected by James A. Sellers, Jun 21 2003

Status

approved