|
| |
|
|
A007564
|
|
Shifts left when INVERT transform applied thrice.
(Formerly M3556)
|
|
18
|
|
|
|
1, 1, 4, 19, 100, 562, 3304, 20071, 124996, 793774, 5120632, 33463102, 221060008, 1473830308, 9904186192, 67015401391, 456192667396, 3122028222934, 21467769499864, 148246598341018, 1027656663676600, 7148588698592956, 49884553176689584
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+2)*x+1))/(2*m*x) are given by : a(n)=sum(k=0,n,(m+1)^k*N(n,k)) where N(n,k)=1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 24 2003
If y=x*A(x) then 3*y^2-(1+2*x)*y+x=0 and x=y*(1-3*y)/(1-2*y). - Michael Somos, Sep 28 2003
The sequence 0,1,4,19,... with g.f. (1-4*x-sqrt(1-8*x+4*x^2))/(6*x) and has a(n)=sum{k=0..floor((n-1)/2), C(n-1,2k)*C(k)*4^(n-1-2*k)*3^k}. a(n+1)=sum{k=0..floor(n/2), C(n,2*k)*C(k)*4^(n-2*k)*3^k} counts Motzkin paths of length n in which the level steps have 4 colors and the up steps have 3. It is the binomial transform of A107264 and corresponds to the series reversion of x/(1+4*x+3*x^2). - Paul Barry, May 18 2005
The Hankel transform of this sequence is 3^C(n+1,2) . - Philippe DELEHAM, Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 2 colors. Example: a(2)=4 because we have UDUD, UUDD, UBD, and URD, where U=(1,1), D=(1,-1), while B (R) is a blue (red) (2,0)-step. - Emeric Deutsch, May 02 2011
a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=19 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^2 = 9 paths of shape HH, 3 paths of shape HUD, 3 paths of shape UDH, 2 paths of shape UHD, and 1 path of each of the shapes UDUD and UUDD. [Emeric Deutsch, May 02 2011]
|
|
|
REFERENCES
|
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, [http://algo.inria.fr/banderier/Papers/DiscMath99.ps Generating Functions for Generating Trees], Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..100
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 443
N. J. A. Sloane, Transforms
|
|
|
FORMULA
|
G.f.: (1+2*x-sqrt(1-8*x+4*x^2))/(6*x). - Emeric Deutsch, Nov 03 2001
a(0)=1, for n>=1 a(n)=sum(k=0, n, 3^k*N(n, k)) where N(n, k)=1/n*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 24 2003
a(n) = Sum_{k=0..n} A088617(n, k)*3^k*(-2)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1 : a(1)=1, a(n)=-2*a(n-1)+3*sum(i=1, n-1, a(i)*a(n-i)) - Benoit Cloitre, Mar 16 2004
a(n) = (4*(2n-1)*a(n-1) - 4*(n-2)*a(n-2)) / (n+1) for n>=2, a(0) = a(1) = 1 . - Philippe DELEHAM, Aug 19 2005
Contribution from Paul Barry, Dec 15 2008: (Start)
G.f.: 1/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x........ (continued fraction).
The g.f. of a(n+1) is 1/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2..... (continued fraction). (End)
a(0) = 1, for n>=1, 3a(n) = A047891(n) [From Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009]
a(n) = upper left term in M^n, M = the production matrix:
1, 1
3, 3, 3
1, 1, 1, 1
3, 3, 3, 3, 3
1, 1, 1, 1, 1, 1
...
- Gary W. Adamson, Jul 08 2011
G.f.: A(x)= (1+2*x-sqrt(1-8*x+4*x^2))/(6*x)= 1/G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ sqrt(6+4*sqrt(3))*(4+2*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012
|
|
|
EXAMPLE
|
1 + x + 4*x^2 + 19*x^3 + 100*x^4 + 562*x^5 + 3304*x^6 + 20071*x^7 + 124996*x^8 + ...
|
|
|
MAPLE
|
A007564_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+3*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A007564_list(21); # Peter Luschny, May 19 2011
|
|
|
MATHEMATICA
|
a[0]=1; a[1]=1; a[n_]/; n>=2 := a[n] = a[n-1] + 3 Sum[a[k-1]a[n-k], {k, n-1}] ; Table[a[n], {n, 0, 10}] [From David Callan, Aug 25 2009]
Table[Hypergeometric2F1[-n, 1 - n, 2, 3], {n, 0, 22}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
|
|
|
PROG
|
(PARI) {a(n) = if( n<1, n==0, sum( k=0, n, 3^k * binomial( n, k) * binomial( n, k+1)) / n)} /* Michael Somos, Sep 28 2003 */
(PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * (1 - 3*x) / (1 - 2*x) + x * O(x^n)), n))} /* Michael Somos, Sep 28 2003 */
|
|
|
CROSSREFS
|
Sequence in context: A211855 A177249 A083882 * A218185 A086624 A151382
Adjacent sequences: A007561 A007562 A007563 * A007565 A007566 A007567
|
|
|
KEYWORD
|
nonn,nice,eigen
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
