cp's OEIS Frontend

A174455 Number of partitions where the number of 1's and 2's are equal.

%I A174455 #49 Aug 13 2025 22:48:16
%S A174455 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 A174455 281,343,428,511,616,759,902,1084,1315,1562,1863,2242,2649,3147,3752,
%U A174455 4424,5222,6190,7266,8545,10062,11776,13782,16157,18832,21964,25622,29777,34589,40200,46556,53912
%N A174455 Number of partitions where the number of 1's and 2's are equal.
%C A174455 From _Omar E. Pol_, Jan 19 2013: (Start)
%C A174455 Column 3 of triangle A220504.
%C A174455 With offset 3, a(n) is also the number of appearances of 3 as the smallest part in all partitions of n.
%C A174455 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 A174455 (End)
%H A174455 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 A174455 G.f.: 1/( (1-x^3) * Product_{n>=3} (1-x^n) ). - _Joerg Arndt_, Jul 07 2012
%F A174455 a(n) = A182713(n+2) - A182713(n) = A240056(n+1) - A240056(n) for n >= 0. - _Clark Kimberling_, Mar 31 2014
%F A174455 a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (9 * 2^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Jan 15 2022
%e A174455 a(8)=9, there are 8 such partitions of 9, they are
%e A174455   #1:    9 =  3* 1 + 3* 2 + 0    + 0    + 0    + 0    + 0    + 0    + 0
%e A174455   #2:    9 =  2* 1 + 2* 2 + 1* 3 + 0    + 0    + 0    + 0    + 0    + 0
%e A174455   #3:    9 =  1* 1 + 1* 2 + 2* 3 + 0    + 0    + 0    + 0    + 0    + 0
%e A174455   #4:    9 =  0    + 0    + 3* 3 + 0    + 0    + 0    + 0    + 0    + 0
%e A174455   #5:    9 =  0    + 0    + 0    + 1* 4 + 1* 5 + 0    + 0    + 0    + 0
%e A174455   #6:    9 =  1* 1 + 1* 2 + 0    + 0    + 0    + 1* 6 + 0    + 0    + 0
%e A174455   #7:    9 =  0    + 0    + 1* 3 + 0    + 0    + 1* 6 + 0    + 0    + 0
%e A174455   #8:    9 =  0    + 0    + 0    + 0    + 0    + 0    + 0    + 0    + 1* 9
%p A174455 b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
%p A174455       else `if`(i=3 and irem(n, 3, 'r')=0, r, 0); for j from 0
%p A174455       to n/i do %+b(n-i*j, i-1) od; % fi
%p A174455     end:
%p A174455 a:= n-> b(n+3, n+3):
%p A174455 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 20 2013
%t A174455 (* See A240056. - _Clark Kimberling_, Mar 31 2014 *)
%t A174455 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 A174455 (PARI)
%o A174455 N=66;  x='x+O('x^N);
%o A174455 gf=1/( (1-x^3) * prod(n=3,N, 1-x^n) );
%o A174455 Vec(gf)
%o A174455 /* _Joerg Arndt_, Jul 07 2012 */
%Y A174455 Cf. A008483, A182713, A240056.
%K A174455 nonn
%O A174455 0,4
%A A174455 _Joerg Arndt_, Nov 28 2010