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A276055
Number of palindromic compositions of n with parts in {1,2,4,6,8,10,...}.
2
1, 1, 2, 1, 4, 2, 7, 3, 13, 6, 23, 10, 42, 19, 75, 33, 136, 61, 244, 108, 441, 197, 793, 352, 1431, 638, 2576, 1145, 4645, 2069, 8366, 3721, 15080, 6714, 27167, 12087, 48961, 21794, 88215, 39254, 158970, 70755, 286439, 127469, 516164, 229725, 930072
OFFSET
0,3
REFERENCES
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
LINKS
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
FORMULA
G.f.: g(z) =(1+z^2 )*(1+z-z^3)/(1-z^2-2z^4+z^6). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, we have g(z)=(1+F(z))/(1-F(z^2)) (see Theorem 1.2 in the Hoggatt et al. reference).
a(2n) = |A078038(n)|. a(2n+1)= A028495(n). - R. J. Mathar, Jan 13 2023
EXAMPLE
a(6) = 7 because the palindromic compositions of 6 with parts in {1,2,4,6,8,...} are 6, 141, 222, 2112, 1221, 11211, and 111111.
MAPLE
g := (1+z^2)*(1+z-z^3)/(1-z^2-2*z^4+z^6): gser:= series(g, z=0, 55): seq(coeff(gser, z, n), n=0..50);
MATHEMATICA
CoefficientList[Series[(1 + x^2) (1 + x - x^3)/(1 - x^2 - 2 x^4 + x^6), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2016 *)
LinearRecurrence[{0, 1, 0, 2, 0, -1}, {1, 1, 2, 1, 4, 2}, 50] (* Harvey P. Dale, Jul 03 2021 *)
CROSSREFS
Sequence in context: A074364 A376318 A256610 * A252866 A008796 A254594
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 17 2016
STATUS
approved