cp's OEIS Frontend

A098123 Number of compositions of n with equal number of even and odd parts.

%I A098123 #25 May 01 2025 08:31:54
%S A098123 1,0,0,2,0,4,6,6,24,28,60,130,190,432,770,1386,2856,5056,9828,18918,
%T A098123 34908,68132,128502,244090,470646,890628,1709136,3271866,6238986,
%U A098123 11986288,22925630,43932906,84349336,161625288,310404768,596009494
%N A098123 Number of compositions of n with equal number of even and odd parts.
%H A098123 Alois P. Heinz, <a href="/A098123/b098123.txt">Table of n, a(n) for n = 0..1000</a>
%F A098123 a(n) = Sum_{k=floor(n/3)..floor(n/2)} C(2*n-4*k,n-2*k)*C(n-1-k,2*n-4*k-1).
%F A098123 Recurrence: n*(2*n-7)*a(n) = 2*(n-2)*(2*n-5)*a(n-2) + 2*(2*n-7)*(2*n-3)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - _Vaclav Kotesovec_, May 01 2014
%F A098123 a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.94696532812840456026081823863... is the root of the equation 1-4*d-2*d^2+d^4 = 0, c = 0.225563290820392765554898545739... is the root of the equation 43*c^4-18*c^2+8*c-1=0. - _Vaclav Kotesovec_, May 01 2014
%F A098123 G.f.: 1/sqrt(1 - 4*x^3/(1-x^2)^2). - _Seiichi Manyama_, May 01 2025
%e A098123 From _Gus Wiseman_, Jun 26 2022: (Start)
%e A098123 The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):
%e A098123   ()  .  .  (12)  .  (14)  (1122)  (16)
%e A098123             (21)     (23)  (1212)  (25)
%e A098123                      (32)  (1221)  (34)
%e A098123                      (41)  (2112)  (43)
%e A098123                            (2121)  (52)
%e A098123                            (2211)  (61)
%e A098123 (End)
%t A098123 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Count[#,_?EvenQ]==Count[#,_?OddQ]&]],{n,0,15}] (* _Gus Wiseman_, Jun 26 2022 *)
%Y A098123 For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).
%Y A098123 Column k=0 of A242498.
%Y A098123 Without multiplicity: A242821, for partitions A241638 (ranked by A325700).
%Y A098123 These compositions are ranked by A355321.
%Y A098123 A047993 counts balanced partitions, ranked by A106529.
%Y A098123 A108950/A108949 count partitions with more odd/even parts.
%Y A098123 A130780/A171966 count partitions with more or as many odd/even parts.
%Y A098123 Cf. A000700, A000712, A001405, A026424, A027193, A028260, A097613, A240009, A277579 (ranked by A349157).
%Y A098123 Cf. A025178.
%K A098123 easy,nonn
%O A098123 0,4
%A A098123 _Vladeta Jovovic_, Sep 24 2004