login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A239631 Triangular array read by rows.  T(n,k) is the number of parts equal to k over all palindromic compositions of n, n>=1, 1<=k<=n. 0
1, 2, 1, 3, 0, 1, 6, 3, 0, 1, 8, 2, 1, 0, 1, 16, 8, 2, 1, 0, 1, 20, 6, 4, 0, 1, 0, 1, 40, 20, 6, 4, 0, 1, 0, 1, 48, 16, 10, 2, 2, 0, 1, 0, 1, 96, 48, 16, 10, 2, 2, 0, 1, 0, 1, 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1, 224, 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums = A239632(n).

LINKS

Table of n, a(n) for n=1..78.

P. Z. Chinn, R. Grimaldi, and S. Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combinatoria 69 (2003), 65-78.

FORMULA

Explicit formula for T(n,k) given in reference [Chinn, Grimaldi, Heubach] as Theorem 6:

a(n) = 0 if n<k or if k<n<2k and n!=k (mod 2);

a(n) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k) if n>=2k and n!=k (mod 2);

a(n) = 1 if n=k;

a(n) = 2^((n-k)/2-1) if k<n<2k and n==k (mod 2);

a(n) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k + 2^floor((k+1)/2-1)) if n>=2k and n==k (mod 2). (end)

O.g.f. for column k: x^k/(1-F(x^2)) + 2*x^(2*k)*(1 + F(x))/(1 - F(x^2))^2 where F(x)= x/(1-x).

EXAMPLE

1,

2,   1,

3,   0,  1,

6,   3,  0,  1,

8,   2,  1,  0, 1,

16,  8,  2,  1, 0, 1,

20,  6,  4,  0, 1, 0, 1,

40,  20, 6,  4, 0, 1, 0, 1,

48,  16, 10, 2, 2, 0, 1, 0, 1,

96,  48, 16, 10,2, 2, 0, 1, 0, 1,

112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1

In the palindromic compositions of 5: 5, 1+3+1, 2+1+2, 1+1+1+1+1  there are T(5,1)=8 ones, T(5,2)=2 twos, and T(5,3)=1 three and T(5,5)=1 five.

MATHEMATICA

nn=15; Table[Take[Drop[Transpose[Map[PadRight[#, nn+1]&, Level[Table[r=Solve[p==1/(1-x)-x^n+y x^n+(x^2/(1-x^2)-x^(2n)+y^2x^(2n))p, p]; CoefficientList[Series[D[p/.r, y]/.y->1, {x, 0, nn}], x], {n, 1, nn}], {2}]]], 1][[n]], n], {n, 1, nn}]//Grid

CROSSREFS

Cf. A016116, A078836, A078836, A079861, A079862, A079863.

Sequence in context: A137587 A168021 A137639 * A113288 A199580 A035215

Adjacent sequences:  A239628 A239629 A239630 * A239632 A239633 A239634

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Mar 22 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 15 12:22 EDT 2019. Contains 327078 sequences. (Running on oeis4.)