OFFSET
0,3
COMMENTS
[1, 3, 12, 52, 232, ...] is INVERT transform of [1, 2, 27, 108, 440, ...] A026726. - Michael Somos, Apr 15 2012
HANKEL transform of sequence and the sequence omitting a(0) is the odd and even bisections of Fibonacci numbers respectively. This is the unique sequence with that property. - Michael Somos, Apr 15 2012
Bisection (even part) of A224747. - Alois P. Heinz, Jul 29 2013
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cyril Banderier, Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
A. Burstein, Restricted Dumont permutations, arXiv:math/0402378 [math.CO], 2004
A. Burstein, Restricted Dumont permutations, Annals of Combinatorics, 9, 2005, 269-280 (Theorem 3.12).
FORMULA
G.f.: [2-(1+x)C(x)]/[2-x-(1+x)C(x)], where C(x)=(1-sqrt(1-4x))/(2x) is the Catalan function.
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) = upper left term in M^n, where M is an infinite square production matrix in which two columns of (1,2,3,...) are prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
1, 1, 0, 0, 0, 0, ...
2, 2, 1, 0, 0, 0, ...
3, 3, 1, 1, 0, 0, ...
4, 4, 1, 1, 1, 0, ...
5, 5, 1, 1, 1, 1, ...
... (End)
Given g.f. A(x), then 0 = A(x)^2 * (x^3 - 2*x^2 + 5*x - 1) + A(x) *(x^2 - 9*x + 2) + (x^2 + 4*x -1). - Michael Somos, Jan 14 2014
0 = a(n)*(16*a(n+1) +6*a(n+2) -14*a(n+3) +210*a(n+4) -128*a(n+5) +18*a(n+6)) +a(n+1)*(-46*a(n+1) +143*a(n+2) -173*a(n+3) -283*a(n+4) +202*a(n+5) -29*a(n+6)) +a(n+2)*(-63*a(n+2) +386*a(n+3) +765*a(n+4) -529*a(n+5) +75*a(n+6)) +a(n+3)*(-559*a(n+3) +509*a(n+4) -149*a(n+5) +19*a(n+6)) +a(n+4)*(-108*a(n+4) +71*a(n+5) -12*a(n+6)) +a(n+5)*(-4*a(n+5) +a(n+6)). - Michael Somos, Jan 14 2014
G.f.: ( 2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x) ) / (2 - 10*x + 4*x^2 - 2*x^3). - Michael Somos, Apr 15 2012
G.f. = (1 - 3*y + y^2) / (1 - 4*y + 3*y^2 - y^3) = 1 / (1 - y / (1 - y / (1 - 2*y / (1 + y / (2 - y))))) where y = (1 - sqrt(1 - 4*x)) / 2. - Michael Somos, Apr 12 2012
D-finite with recurrence (-n+1)*a(n) +4*(2*n-3)*a(n-1) +(-13*n+19)*a(n-2) +(-13*n+75)*a(n-3) +(5*n-29)*a(n-4) +2*(-2*n+9)*a(n-5)=0. - R. J. Mathar, Jul 27 2013
EXAMPLE
G.f. = 1 + x + 3*x^2 + 12*x^3 + 52*x^4 + 232*x^5 + 1049*x^6 + 4777*x^7 + 21845*x^8 + ...
MAPLE
C:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*C)/(2-x-(1+x)*C): Gser:=series(G, x=0, 30): seq(coeff(Gser, x, n), n=0..26);
MATHEMATICA
a[ n_] := SeriesCoefficient[ (2 - 9 x + x^2 + (x + x^2) Sqrt[1 - 4 x]) / (2 (1 - 5 x + 2 x^2 - x^3)), {x, 0, n}]; (* Michael Somos, Jan 14 2014 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( (2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x + x * O(x^n)) ) / (2 * (1 - 5*x + 2*x^2 - x^3)), n))}; /* Michael Somos, Jan 14 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 19 2006
STATUS
approved