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A354786
Irregular triangle read by rows: T(n,k) is the number of anti-palindromic compositions of n of length k, n >= 0, 0 <= k <= floor((2*n+1)/3).
2
1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 4, 4, 0, 1, 4, 8, 4, 0, 1, 6, 12, 8, 4, 0, 1, 6, 18, 20, 12, 0, 1, 8, 24, 32, 32, 8, 0, 1, 8, 32, 56, 64, 24, 8, 0, 1, 10, 40, 80, 120, 72, 32, 0, 1, 10, 50, 120, 200, 152, 104, 16, 0, 1, 12, 60, 160, 320, 312, 256, 64, 16
OFFSET
0,8
COMMENTS
A composition S with sum n and length k is anti-palindromic if S(i) != S(k+1-i) for 1 <= i < floor(k). - Andrew Howroyd, Feb 28 2023
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..3467 (rows 0..100)
George E. Andrews, Matthew Just, and Greg Simay, Anti-palindromic compositions, arXiv:2102.01613 [math.CO], 2021. Also Fib. Q., 60:2 (2022), 164-176. See Table 2.
FORMULA
G.f.: A(x,y) = (1 + x*y/(1 - x))/(1 - 2*x^3*y^2/((1 + x)*(1 - x)^2)). - Andrew Howroyd, Feb 28 2023
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1,
0, 1, 2;
0, 1, 2, 2;
0, 1, 4, 4;
0, 1, 4, 8, 4;
0, 1, 6, 12, 8, 4;
0, 1, 6, 18, 20, 12;
...
PROG
(PARI) T(n)=[Vecrev(p) | p<-Vec((1 + x*y/(1 - x))/(1 - 2*x^3*y^2/((1 + x)*(1 - x)^2)) + O(x*x^n))]
{ my(rows=T(12)); for(i=1, #rows, print(rows[i])) } \\ Andrew Howroyd, Feb 28 2023
CROSSREFS
Row sums are Tribonacci numbers (A000213).
Cf. A354787.
Sequence in context: A035465 A096144 A118401 * A147767 A113678 A305436
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 13 2022
EXTENSIONS
Terms a(33) and beyond from Andrew Howroyd, Feb 28 2023
STATUS
approved