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A119749
Number of compositions of n into odd blocks with one element in each block distinguished.
2
1, 1, 4, 7, 15, 32, 65, 137, 284, 591, 1231, 2560, 5329, 11089, 23076, 48023, 99935, 207968, 432785, 900633, 1874236, 3900319, 8116639, 16890880, 35150241, 73148321, 152223044, 316779047, 659223215, 1371856032, 2854858465
OFFSET
1,3
COMMENTS
The sequence is the INVERT transform of the aerated odd integers. - Gary W. Adamson, Feb 02 2014
Number of compositions of n into odd parts where there is 1 sort of part 1, 3 sorts of part 3, 5 sorts of part 5, ... , 2*k-1 sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014
LINKS
R. X. F. Chen and L. W. Shapiro, On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2), J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).
Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).
B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.
FORMULA
G.f.: (x+x^3)/(x^4 - x^3 -2x^2 -x +1).
a(n) = A092886(n)+A092886(n-2). - R. J. Mathar, Mar 08 2018
Sum_{k=0..n} a(k) = (3*a(n) + 2*a(n-1) - a(n-3))/2 - 1. - Xilin Wang and Greg Dresden, Aug 27 2020
EXAMPLE
a(3) = 4 since Abc, aBc, abC come from one block of size 3 and A/B/C comes from having three blocks. The capital letters are the distinguished elements.
MATHEMATICA
Rest@ CoefficientList[ Series[x(1 + x^2)/(x^4 - x^3 - 2x^2 - x + 1), {x, 0, 50}], x] (* Robert G. Wilson v *)
CROSSREFS
Cf. A105309, A052530, A000045, A030267. Row sums of A292835.
Sequence in context: A131090 A178615 A131935 * A201498 A145970 A232048
KEYWORD
easy,nonn
AUTHOR
Louis Shapiro, Jul 30 2006
STATUS
approved