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A275592 Number of compositions of n if only the order of the even numbers matter. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

EXAMPLE

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).

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 *)

CROSSREFS

Cf. A000009, A011782, A275548.

Sequence in context: A266775 A024790 A308271 * A319635 A179822 A319769

Adjacent sequences:  A275589 A275590 A275591 * A275593 A275594 A275595

KEYWORD

nonn

AUTHOR

Gregory L. Simay, Aug 02 2016

STATUS

approved

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Last modified September 20 16:05 EDT 2020. Contains 337265 sequences. (Running on oeis4.)