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URL: https://oeis.org/A083751

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A083751
Number of partitions of n into >= 2 parts and with minimum part >= 2.
18
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
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
First differences of A000094.
Sequence in context: A211541 A026926 A332557 * A379720 A034401 A345729
KEYWORD
nonn
AUTHOR
Jon Perry, Jun 17 2003
EXTENSIONS
More terms from Vladeta Jovovic and Wouter Meeussen, Jun 18 2003
Description corrected by James Sellers, Jun 21 2003
STATUS
approved