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A275592 Number of compositions of n if only the order of the even numbers matter.

%I A275592 #21 Jun 02 2018 08:58:03
%S A275592 1,1,2,3,5,7,12,16,26,35,56,74,117,154,241,317,492,645,998,1306,2014,
%T A275592 2634,4053,5296,8139,10630,16321,21310,32699,42684,65472,85452,131038,
%U A275592 171012,262198,342161,524552,684497,1049300,1369216,2098849,2738710,4198011
%N A275592 Number of compositions of n if only the order of the even numbers matter.
%C A275592 The number of compositions of n = 2k with only even numbers is c(k) = A011782(k). The number of partitions of n with only odd numbers is the strict partition q(n) = A000009(n). Enumerating a(n) is therefore a sum of products of composition numbers and strict partition numbers. (See formulas.)
%H A275592 Alois P. Heinz, <a href="/A275592/b275592.txt">Table of n, a(n) for n = 0..1000</a>
%F A275592 a(2k+1) = Sum_{j=0..k} c(j)*q(2k+1-2j), where c(j) = A011782(j), the number of compositions of j, and q(j) = A000009(j), the number of strict partitions of j.
%F A275592 a(2k) = Sum_{j=0..k} c(j)*q(2k - 2j).
%F A275592 a(n) = 2*a(n-2) + q(n) - q(n-2).
%F A275592 G.f.: (1 - x^2)/(1 - 2*x^2) * Product_{n>=1} (1 + x^n). - _Peter Bala_, Aug 03 2016
%F A275592 a(n) ~ c * 2^(n/2), where c = (QPochhammer[-1, 1/sqrt(2)] + (-1)^n*QPochhammer[-1, -1/sqrt(2)])/8, c = 2.002012668882683075956932277149607919866122388... if n is even and c = 1.8471591618236152130512812517147483461076894... if n is odd. - _Vaclav Kotesovec_, Jun 02 2018
%e A275592 The compositions enumerated by a(6) = 12 are (6), (5,1)=(1,5), (4,2), (2,4), (3,3), (4,1,1)=(1,4,1)=(1,1,4), (3,2,1)=(1,2,3)=(2,3,1)=(2,1,3)=(3,1,2)=(1,3,2), (2,2,2), (3,1,1,1)=(1,3,1,1)=(1,1,3,1)=(1,1,1,3), (2,2,1,1)=(2,1,2,1)=(2,1,1,2)=(1,2,1,2)=(1,1,2,2)=(1,2,2,1), (2,1,1,1,1)=(1,2,1,1,1)=(1,1,2,1,1,)=(1,1,1,2,1)=(1,1,1,1,2), (1,1,1,1,1,1).
%e A275592 The compositions enumerated by a(5) = 7 are (5), (4,1)=(1,4), (3,2)=(2,3), (3,1,1)=(1,3,1)=(1,1,3), (2,2,1)=(2,1,2)=(1,2,2), (2,1,1,1)=(1,2,1,1)=(1,1,2,1)=(1,1,1,2), (1,1,1,1,1).
%p A275592 b:= proc(n, i, p) option remember; (t-> `if`(n=0, p!,
%p A275592       `if`(i<1, 0, add(b(n-i*j, i-1, p+`if`(t, j, 0))/
%p A275592       `if`(t, j, 0)!, j=0..n/i))))(i::even)
%p A275592     end:
%p A275592 a:= n-> b(n$2, 0):
%p A275592 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 03 2016
%t A275592 nmax = 40; CoefficientList[Series[(1 - x^2)/(1 - 2*x^2)*Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 02 2018 *)
%Y A275592 Cf. A000009, A011782, A275548.
%K A275592 nonn
%O A275592 0,3
%A A275592 _Gregory L. Simay_, Aug 02 2016