A318947 Column 2 of triangle A318945.
0, 0, 0, 0, 1, 6, 26, 97, 331, 1064, 3277, 9775, 28448, 81201, 228211, 633384, 1740037, 4740327, 12825008, 34500649, 92372683, 246352952, 654878173, 1736172895, 4592568896, 12125944161, 31967715811, 84170419272, 221388694261, 581807602839, 1527909651152, 4010192518105
Offset: 0
Keywords
Links
- Muniru A Asiru, Table of n, a(n) for n = 0..2000
- Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L., Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Mathematics (2018), 341(10), 2789-2807.
Crossrefs
Cf. A318945.
Programs
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GAP
Concatenation([0,0,0],List([3..31],n->Fibonacci(2*n)-(n^2+9*n+28)*2^(n-6))); # Muniru A Asiru, Oct 28 2018
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Maple
a := n -> `if`(n < 3, 0, combinat:-fibonacci(2*n) - (n^2 + 9*n + 28)*2^(n - 6)): seq(a(n), n=0..31); # Peter Luschny, Oct 28 2018
Formula
Let alpha(n) = Sum_{k=0..n} binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1) then alpha(n) = a(n+3) for n >= 0. - Peter Luschny, Oct 28 2018
Conjectures from Colin Barker, Oct 28 2018: (Start)
G.f.: x^4*(1 - x)^3 / ((1 - 2*x)^3*(1 - 3*x + x^2)).
a(n) = 9*a(n-1) - 31*a(n-2) + 50*a(n-3) - 36*a(n-4) + 8*a(n-5) for n>7. (End)
Extensions
More terms from Peter Luschny, Oct 28 2018
a(30) corrected by Muniru A Asiru, Oct 28 2018