A079977 -id:A079977 - cp's OEIS frontend

A239366 Triangular array read by rows: T(n,k) is the number of palindromic compositions of n having exactly k 1's, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 0, 3, 0, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 1, 5, 0, 5, 0, 4, 0, 1, 0, 1, 3, 2, 3, 2, 1, 3, 1, 0, 0, 1, 8, 0, 10, 0, 7, 0, 5, 0, 1, 0, 1, 5, 3, 5, 5, 4, 3, 1, 4, 1, 0, 0, 1, 13, 0, 18, 0, 16, 0, 9, 0, 6, 0, 1, 0, 1, 8, 5, 10, 8, 7, 9, 5, 4, 1, 5, 1, 0, 0, 1

Offset: 0

Geoffrey Critzer, Mar 20 2014

Row sums = 2^floor(n/2).

T(n,0) = A053602(n-1) for n>0, T(n,1) = A079977(n-5) for n>4, T(2n+1,3) = A006367(n-1) for n>0, both bisections of column k=2 contain A010049. - Alois P. Heinz, Mar 21 2014

			1,
0, 1,
1, 0, 1,
1, 0, 0, 1,
2, 0, 1, 0, 1,
1, 1, 1, 0, 0, 1,
3, 0, 3, 0, 1, 0, 1,
2, 1, 1, 2, 1, 0, 0, 1,
5, 0, 5, 0, 4, 0, 1, 0, 1,
3, 2, 3, 2, 1, 3, 1, 0, 0, 1
There are eight palindromic compositions of 6: T(6,0)=3 because we have: 6, 3+3, 2+2+2.  T(6,2)=3 because we have: 1+4+1, 2+1+1+2, 1+2+2+1.  T(6,4)=1 because we have: 1+1+2+1+1. T(6,6)=1 because we have: 1+1+1+1+1+1.

Alois P. Heinz, Rows n = 0..140, flattened

b:= proc(n) option remember;  `if`(n=0, 1, expand(
      add(b(n-j)*`if`(j=1, x^2, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))
    (add(b(i)*`if`(n-2*i=1, x, 1), i=0..n/2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Mar 21 2014

nn=15;Table[Take[CoefficientList[Series[((1+x)*(1-x+x^2+x*y-x^2*y))/(1-x^2-x^4-x^2*y^2+x^4*y^2),{x,0,nn}],{x,y}][[n]],n],{n,1,nn}]//Grid

G.f.: G(x,y) = ((1 + x)*(1 - x + x^2 + x*y - x^2*y))/(1 - x^2 - x^4 - x^2*y^2 + x^4*y^2). Satisfies G(x,y) = 1/(1 - x) - x + y*x + (x^2/(1 - x^2) - x^2 +y^2*x^2)*G(x,y).

A336749 Number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 4.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 3, 6, 5, 10, 12, 14, 25, 27, 40, 57, 68, 104, 133, 177, 255, 324, 454, 617, 811, 1136, 1507, 2042, 2803, 3729, 5109, 6904, 9290, 12692, 17070, 23152, 31430, 42361, 57567, 77842, 105279, 142865, 193040, 261589, 354316, 479189, 649498, 878905

Offset: 5

Ethan Patrick White, Aug 02 2020

Permutations in which adjacent terms sum to a particular value is a property central to the sequences A090460, A071984, A108658, A272259, and A107929.

Ethan P. White, Richard K. Guy, Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020. See Table A.1 p. 31.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,2,2,1,1,1).

See A079977 or A017899 for other sequences counting similar circular arrangements of positive integers.

CoefficientList[ Series[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 + x - x^3 - x^4 - 2*x^5 - 2*x^6 - x^7 - x^8 - x^9), {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 07 2020 *)

a(n) = -a(n-1) + a(n-3) + a(n-4) + 2*a(n-5) + 2*a(n-6) + a(n-7) + a(n-8) + a(n-9) for n > 13.

G.f.: x^5*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 + x - x^3 - x^4 - 2*x^5 - 2*x^6 - x^7 - x^8 - x^9). - Stefano Spezia, Aug 03 2020

cp's OEIS Frontend

A239366 Triangular array read by rows: T(n,k) is the number of palindromic compositions of n having exactly k 1's, n>=0, 0<=k<=n.

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