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%I #46 Jan 15 2022 03:08:45
%S 1,0,0,2,1,1,4,3,4,8,8,10,17,18,23,34,39,48,67,78,97,127,151,185,237,
%T 281,343,428,511,616,759,902,1084,1315,1562,1863,2242,2649,3147,3752,
%U 4424,5222,6190,7266,8545,10062,11776,13782,16157,18832,21964,25622,29777,34589,40200,46556,53912
%N Number of partitions where the number of 1's and 2's are equal.
%C From _Omar E. Pol_, Jan 19 2013: (Start)
%C Column 3 of triangle A220504.
%C With offset 3, a(n) is also the number of appearances of 3 as the smallest part in all partitions of n.
%C Also consider the sequence formed by [0, 0] together with this sequence, with offset 1, then it appears that A027336(n) = Sum_{j=1..3} a(n+j), n >= 0.
%C (End)
%H Seiichi Manyama, <a href="/A174455/b174455.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F G.f.: 1/( (1-x^3) * Product_{n>=3} (1-x^n) ). [_Joerg Arndt_, Jul 07 2012]
%F a(n) = A182713(n+2) - A182713(n) = A240056(n+1) - A240056(n) for n >= 0. - _Clark Kimberling_, Mar 31 2014
%F a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (9 * 2^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Jan 15 2022
%e a(8)=9, there are 8 such partitions of 9, they are
%e #1: 9 = 3* 1 + 3* 2 + 0 + 0 + 0 + 0 + 0 + 0 + 0
%e #2: 9 = 2* 1 + 2* 2 + 1* 3 + 0 + 0 + 0 + 0 + 0 + 0
%e #3: 9 = 1* 1 + 1* 2 + 2* 3 + 0 + 0 + 0 + 0 + 0 + 0
%e #4: 9 = 0 + 0 + 3* 3 + 0 + 0 + 0 + 0 + 0 + 0
%e #5: 9 = 0 + 0 + 0 + 1* 4 + 1* 5 + 0 + 0 + 0 + 0
%e #6: 9 = 1* 1 + 1* 2 + 0 + 0 + 0 + 1* 6 + 0 + 0 + 0
%e #7: 9 = 0 + 0 + 1* 3 + 0 + 0 + 1* 6 + 0 + 0 + 0
%e #8: 9 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1* 9
%p b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
%p else `if`(i=3 and irem(n, 3, 'r')=0, r, 0); for j from 0
%p to n/i do %+b(n-i*j, i-1) od; % fi
%p end:
%p a:= n-> b(n+3, n+3):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 20 2013
%t (See A240056.) - _Clark Kimberling_, Mar 31 2014
%t m = 66; gf = 1/((1-x^3)*Product[1-x^n, {n, 3, m}]) + O[x]^m; CoefficientList[gf, x] (* _Jean-François Alcover_, Jul 02 2015, after _Joerg Arndt_ *)
%o (PARI)
%o N=66; x='x+O('x^N);
%o gf=1/( (1-x^3) * prod(n=3,N, 1-x^n) );
%o Vec(gf)
%o /* _Joerg Arndt_, Jul 07 2012 */
%Y Cf. A008483, A182713, A240056.
%K nonn
%O 0,4
%A _Joerg Arndt_, Nov 28 2010
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