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%I #38 May 24 2021 03:15:57
%S 0,0,0,1,1,3,3,6,7,11,13,20,23,33,40,54,65,87,104,136,164,209,252,319,
%T 382,477,573,707,846,1038,1237,1506,1793,2166,2572,3093,3659,4377,
%U 5169,6152,7244,8590,10086,11913,13958,16423,19195,22518,26251,30700,35716
%N Number of partitions of n into >= 2 parts and with minimum part >= 2.
%C 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
%C Also number of partitions of n that contain emergent parts (Cf. A182699). - _Omar E. Pol_, Oct 21 2011
%C 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
%C 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
%H Alois P. Heinz, <a href="/A083751/b083751.txt">Table of n, a(n) for n = 1..1000</a>
%H Robert Schneider, <a href="https://arxiv.org/abs/1912.00575">Nuclear partitions and a formula for p(n)</a>, arXiv:1912.00575 [math.NT], 2019.
%F a(n) = A000041(n) - A000041(n-1) - 1, n > 1. - _Vladeta Jovovic_, Jun 18 2003
%F G.f.: Sum_{j>=2} x^(2j)/Product_{i=1..j} (1-x^i). - _Emeric Deutsch_, Mar 29 2006
%F a(n) = A002865(n) - 1, n > 1. - _Omar E. Pol_, Oct 21 2011
%F a(n) = A187219(n) - 1. - _Omar E. Pol_, Mar 04 2012
%e a(6) = 3, as 6 = 2+4 = 3+3 = 2+2+2.
%e a(6) = 3 because 6 = 2+4 = 3+3 = 2+2+2.
%p 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
%t Drop[CoefficientList[Series[1/Product[(1-x^k)^1, {k, 2, 50}], {x, 0, 50}], x]-1, 2]
%t (* or *) Table[Count[IntegerPartitions[n], q_List /; Length[q] > 1 && Min[q] >= 2 ], {n, 24}]
%Y Cf. A053445, A072380, A008483, A026796, A035989, A036000, A002865, A081094.
%Y First differences of A000094.
%K nonn
%O 1,6
%A _Jon Perry_, Jun 17 2003
%E More terms from _Vladeta Jovovic_ and _Wouter Meeussen_, Jun 18 2003
%E Description corrected by _James A. Sellers_, Jun 21 2003
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