|
|
A064613
|
|
Second binomial transform of the Catalan numbers.
|
|
27
|
|
|
1, 3, 10, 37, 150, 654, 3012, 14445, 71398, 361114, 1859628, 9716194, 51373180, 274352316, 1477635912, 8016865533, 43773564294, 240356635170, 1326359740956, 7351846397334, 40913414754324, 228508350629892
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Exponential convolution of Catalan numbers and powers of 2. - Vladeta Jovovic, Dec 03 2004
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 4 colors. Example: a(3)=37 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 4 paths of shape UHD. - Emeric Deutsch, May 02 2011
a(n) is the number of Schroeder paths of semilength n in which the (2,0)-steps come in 2 colors and having no (2,0)-steps at levels 1,3,5,... - José Luis Ramírez Ramírez, Mar 30 2013
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A126930.)
O.g.f.: G(x) = C[P[P(x,-1),-1]] = C[P(x,-2)] = [1-sqrt(1-4*x/(1-2x)]/2 = x*A064613(x).
Ginv(x) = Pinv[Cinv(x),-2] = P[Cinv(x),2] = x(1-x)/[1+2x(1-x)] = (x-x^2)/[1+2(x-x^2)] = x - 3 x^2 + 8 x^3 - ... is -A155020(-x) ignoring first term there. (Cf. A146559, A125145.)(End)
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1).
a(n) = 2^n*hypergeom([1/2, -n], [2], -2).
With offset 1: a(1) = 1, a(n) = 2^(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence (n+1)*a(n) = (8*n-2)*a(n-1) - (12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - Vladeta Jovovic, Dec 03 2004
G.f.: 1/(1-3*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). - Paul Barry, Jul 02 2009
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
3, 1, 0, 0, ...
1, 3, 1, 0, ...
1, 1, 3, 1, ...
1, 1, 1, 3, ...
... (End)
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020
|
|
MATHEMATICA
|
CoefficientList[Series[(1-Sqrt[(1-6*x)/(1-2*x)])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2013 *)
a[n_] := 2^n Hypergeometric2F1[1/2, -n, 2, -2];
|
|
PROG
|
(PARI) x='x+O('x^66); Vec((1-sqrt((1-6*x)/(1-2*x)))/(2*x)) /* Joerg Arndt, Mar 31 2013 */
(Magma) I:=[3, 10]; [1] cat [n le 2 select I[n] else ((8*n-2)*Self(n-1)-(12*n-12)*Self(n-2))div (n+1): n in [1..30]]; // Vincenzo Librandi, Jan 23 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|