OFFSET
0,15
COMMENTS
A composition S with sum n and length k is a reduced anti-palindromic composition if S(i) < S(k+1-i) for 1 <= i <= floor(k/2). - Andrew Howroyd, Feb 28 2023
A composition S with sum n and length k is an Arndt composition if S(2i-1) > S(2i) for all i >= 1. T(n,k) also counts these compositions. - Daniel Checa, Jan 05 2024
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 3.
Daniel F. Checa and José L. Ramírez, Arndt compositions: a generating functions approach, arXiv:2311.15388 [math.CO], 2023. See also Integers (2024) Vol. 24, A35, p. 4.
Daniel F. Checa, Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations, 2023. p. 17.
FORMULA
G.f.: A(x,y) = (1 + x*y/(1 - x))/(1 - x^3*y^2/((1 + x)*(1 - x)^2)). - Andrew Howroyd, Feb 28 2023
From Daniel Checa, Jan 03 2024: (Start)
G.f. of the k-th column, k >= 1: z^floor(3*k/2)/((1-z)^k*(1+z)^ floor(k/2)).
T(n, k) = Sum_{i=floor(k/2)..n-k} binomial(n-i-1, k-1)*binomial(i-1, floor(k/2) - 1)*(-1)^(i + floor(k/2)) for k >= 2.
(End)
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 1, 2, 2;
0, 1, 2, 4, 1;
0, 1, 3, 6, 2, 1;
0, 1, 3, 9, 5, 3;
...
PROG
(PARI) T(n)=[Vecrev(p) | p<-Vec((1 + x*y/(1 - x))/(1 - 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
(Python)
from math import comb as binomial
def T(n, k):
if k == 0: return k ** n
if k == 1: return 1
return sum(binomial(n - i - 1, k - 1) * binomial(i - 1, k // 2 - 1)
* (-1) ** (i + k // 2) for i in range(k // 2, n - k + 1))
for n in range(11): print([T(n, k) for k in range(1 + (2 * n + 1) // 3)])
# Peter Luschny, Jan 03 2024
CROSSREFS
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