A275592 Number of compositions of n if only the order of the even numbers matter.
1, 1, 2, 3, 5, 7, 12, 16, 26, 35, 56, 74, 117, 154, 241, 317, 492, 645, 998, 1306, 2014, 2634, 4053, 5296, 8139, 10630, 16321, 21310, 32699, 42684, 65472, 85452, 131038, 171012, 262198, 342161, 524552, 684497, 1049300, 1369216, 2098849, 2738710, 4198011
Offset: 0
Keywords
Examples
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). 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).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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Maple
b:= proc(n, i, p) option remember; (t-> `if`(n=0, p!, `if`(i<1, 0, add(b(n-i*j, i-1, p+`if`(t, j, 0))/ `if`(t, j, 0)!, j=0..n/i))))(i::even) end: a:= n-> b(n$2, 0): seq(a(n), n=0..50); # Alois P. Heinz, Aug 03 2016 -
Mathematica
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 *)
Formula
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.
a(2k) = Sum_{j=0..k} c(j)*q(2k - 2j).
a(n) = 2*a(n-2) + q(n) - q(n-2).
G.f.: (1 - x^2)/(1 - 2*x^2) * Product_{n>=1} (1 + x^n). - Peter Bala, Aug 03 2016
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
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