A354787 Irregular triangle read by rows: T(n,k) is the number of reduced anti-palindromic compositions of n of length k, n >= 0, 0 <= k <= floor((2*n+1)/3).

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, 0, 1, 4, 12, 8, 8, 1, 0, 1, 4, 16, 14, 16, 3, 1, 0, 1, 5, 20, 20, 30, 9, 4, 0, 1, 5, 25, 30, 50, 19, 13, 1, 0, 1, 6, 30, 40, 80, 39, 32, 4, 1, 0, 1, 6, 36, 55, 120, 69, 71, 14, 5

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

Row sums are Fibonacci numbers (A000045).

Cf. A354786.

Sequence in context: A029341 A240181 A079070 * A096994 A035370 A306706

Adjacent sequences: A354784 A354785 A354786 * A354788 A354789 A354790

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