cp's OEIS Frontend

A182712 Number of 2's in the last section of the set of partitions of n.

%I A182712 #56 May 16 2020 01:30:44
%S A182712 0,0,1,0,2,1,4,3,8,7,15,15,27,29,48,53,82,94,137,160,225,265,362,430,
%T A182712 572,683,892,1066,1370,1640,2078,2487,3117,3725,4624,5519,6791,8092,
%U A182712 9885,11752,14263,16922,20416,24167,29007,34254,40921,48213,57345,67409
%N A182712 Number of 2's in the last section of the set of partitions of n.
%C A182712 Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
%C A182712 Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
%C A182712 Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - _Omar E. Pol_, Nov 13 2011
%C A182712 Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - _Omar E. Pol_, Dec 01 2013
%H A182712 Alois P. Heinz, <a href="/A182712/b182712.txt">Table of n, a(n) for n = 0..1000</a>
%F A182712 It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - _Omar E. Pol_, Feb 04 2012
%F A182712 G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Jan 03 2017
%F A182712 a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Jun 02 2018
%e A182712 a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
%e A182712 From _Omar E. Pol_, Oct 27 2012: (Start)
%e A182712 ----------------------------------
%e A182712 Last section               Number
%e A182712 of the set of                of
%e A182712 partitions of 6             2's
%e A182712 ----------------------------------
%e A182712 6 .......................... 0
%e A182712 3 + 3 ...................... 0
%e A182712 4 + 2 ...................... 1
%e A182712 2 + 2 + 2 .................. 3
%e A182712 .   1 ...................... 0
%e A182712 .       1 .................. 0
%e A182712 .       1 .................. 0
%e A182712 .           1 .............. 0
%e A182712 .           1 .............. 0
%e A182712 .               1 .......... 0
%e A182712 .                   1 ...... 0
%e A182712 ---------------------------------
%e A182712 .   8 - 4 =                  4
%e A182712 .
%e A182712 In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
%e A182712 (End)
%t A182712 Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* _Robert Price_, May 15 2020 *)
%o A182712 (Sage) A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # _Omar E. Pol_, Nov 13 2011
%Y A182712 Column 2 of A194812.
%Y A182712 Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.
%K A182712 nonn,easy
%O A182712 0,5
%A A182712 _Omar E. Pol_, Nov 28 2010