OFFSET
0,2
COMMENTS
For n >= 2, a(n) equals 2^(2n+1) times the coefficient of Pi in 2F1([3/2, n+1], [5/2], -1). - John M. Campbell, Jul 17 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243 [math.CO], 2012. - From N. J. A. Sloane, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, arXiv:1302.2274 [math.CO], 2013.
Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, 15 (2015), #A16.
FORMULA
G.f.: (5 - 2*x - 3*sqrt(1-4*x))/2. - G. C. Greubel, Feb 18 2019
EXAMPLE
A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 + ...
A(x)^(1/2) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + ...
MAPLE
A120589List := proc(m) local A, P, n; A := [1, 2, 3]; P := [3];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A120589List(26); # Peter Luschny, Mar 26 2022
MATHEMATICA
Join[{1, 2, 3}, Table[3*(2*n)!/n!/(n+1)!, {n, 2, 40}]]
CoefficientList[Series[(5-2x -3Sqrt[1-4x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 18 2019 *)
PROG
(PARI) {a(n)=local(A=1+x+x^2+x*O(x^n)); for(i=0, n, A=A-3*A+2+x+A^2); polcoeff(A^2, n)}
(PARI) my(x='x+O('x^30)); Vec((5-2*x-3*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 18 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-2*x-3*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 18 2019
(Sage) ((5-2*x-3*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
STATUS
approved