A054444 Even-indexed terms of A001629(n), n >= 2, (Fibonacci convolution).

1, 5, 20, 71, 235, 744, 2285, 6865, 20284, 59155, 170711, 488400, 1387225, 3916061, 10996580, 30737759, 85573315, 237387960, 656451269, 1810142185, 4978643596, 13661617195, 37409025455, 102238082976, 278920277425, 759695287349

Offset: 0

Author

Wolfdieter Lang, Apr 07 2000

Keywords

• easy
• nonn

Comments

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

Links

• Jinyuan Wang, Table of n, a(n) for n = 0..1000
• Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.
• Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
• Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 5, 17, 19.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.

Crossrefs

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

Programs

PARI

a(n) = ((2*n+1)*fibonacci(2*(n+1))+4*(n+1)*fibonacci(2*n+1))/5; \\ Jinyuan Wang, Jul 28 2019

Formula

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

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

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

a(n) = Sum_{k=1..n+1} k*binomial(2*n-2*k+2, k). - Emeric Deutsch, Jun 11 2003

a(n) ~ n*(3 + sqrt(5))^(1+n)*2^(-n)/5. - Stefano Spezia, Mar 29 2022

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