A054444 - OEIS

%I #47 Mar 04 2025 17:21:49

%S 1,5,20,71,235,744,2285,6865,20284,59155,170711,488400,1387225,

%T 3916061,10996580,30737759,85573315,237387960,656451269,1810142185,

%U 4978643596,13661617195,37409025455,102238082976,278920277425,759695287349

%N Even-indexed terms of A001629(n), n >= 2, (Fibonacci convolution).

%C 8*a(n) is the number of Boolean (equivalently, lattice, modular lattice, distributive lattice) intervals of the form [s,w] in the Bruhat order on S_n, where s is a simple reflection. - _Bridget Tenner_, Jan 16 2020

%H Jinyuan Wang, <a href="/A054444/b054444.txt">Table of n, a(n) for n = 0..1000</a>

%H Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/2203.13205">Honeycombs in the Pascal triangle and beyond</a>, arXiv:2203.13205 [math.HO], 2022. See p. 5.

%H Éva Czabarka, Rigoberto Flórez, and Leandro Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.

%H Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 5, 17, 19.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H B. E. Tenner, <a href="https://arxiv.org/abs/2001.05011">Interval structures in the Bruhat and weak orders</a>, arXiv:2001.05011 [math.CO], 2020.

%F a(n) = ((2*n+1)*F(2*(n+1)) + 4*(n+1)*F(2*n+1))/5, with F(n) = A000045(n) (Fibonacci numbers).

%F a(n) = A060920(n+1, 1).

%F G.f.: (1 - x + x^2)/(1 - 3*x + x^2)^2.

%F a(n) = Sum_{k=1..n+1} k*binomial(2*n-2*k+2, k). - _Emeric Deutsch_, Jun 11 2003

%F a(n) ~ n*(3 + sqrt(5))^(1+n)*2^(-n)/5. - _Stefano Spezia_, Mar 29 2022

%F E.g.f.: exp(3*x/2)*(5*(5 + 14*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(7 + 30*x)*sinh(sqrt(5)*x/2))/25. - _Stefano Spezia_, Mar 04 2025

%o (PARI) a(n) = ((2*n+1)*fibonacci(2*(n+1))+4*(n+1)*fibonacci(2*n+1))/5; \\ _Jinyuan Wang_, Jul 28 2019

%Y Cf. A000045, A001629, A001870, A060920, A196265.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_, Apr 07 2000