|
| |
|
|
A060693
|
|
Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ....] where DELTA is the operator defined in A084938.
|
|
20
| |
|
|
1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the second to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g. T(n,k)=C(n,k)C(2n-k,n-1)/n for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
A090181*A007318 as infinite lower triangular matrices . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 14 2008]
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). [From Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009]
|
|
|
REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
|
|
|
LINKS
| G. E. Cossali, A Common Generating Function of Catalan Numbers and Other Integer Sequences, J. Int. Seqs. 6 (2003).
|
|
|
FORMULA
| If C_n(x) is the gf of row n of the Narayana numbers (A001263), C_n(x) = sum({n choose k-1}{n-1 choose k-1}/k x^k, {k,1,n}) and T_n(x) is the gf of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]sum(A001263(n,k) (x+1)^k, (k,1,n)) - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007, Jan 31 2007
G.f.: (1-ty-sqrt((1-yt)^2-4y))/2.
T(n, k) = C(2n-k, n)*C(n, k)/(n-k+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 07 2003
A060693(n, k) = C(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 30 2003
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x=-1,0,1,2,3,4,5,6,7,8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 01 2007
T(n,k)=Sum_[j, j>=0}A090181(n,j)*binomial(j,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 04 2007
Sum_[k, 0<=k<=n}T(n,k)*x^(n-k)= (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x= -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 29 2009: (Start)
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k)=[k<=n]*sum{j=0..n, C(n,j)^2*C(j,k)}/(n-k+1). [From Paul Barry (pbarry(AT)wit.ie), May 28 2009]
T(n,k) = A104684(n,k)/(n-k+1). - Peter Luschny, May 17 2011
Contribution from Tom Copeland, Sept 21 2011: (Start)
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx)=(1+t+(2+t)*x+x^2)^2 /(1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*[(H(x,t)*d/dx)^n] x evaluated at x=0, i.e., F(x,t) = exp[x*H(u,t)*d/du] u, evaluated at u = 0.
Also, dF(x,t)/dx = H(F(x,t),t). (End)
|
|
|
EXAMPLE
| {1}, {1,1}, {2,3,1}, {5,10,6,1}, {14,35,30,10,1}, ...
|
|
|
MAPLE
| A060693 := (n, k) -> binomial(n, k)*binomial(2*n-k, n)/(n-k+1); - Peter Luschny, May 17 2011
|
|
|
MATHEMATICA
| t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)
|
|
|
CROSSREFS
| Cf. A006318, A000108, A001700.
Triangle in A088617 transposed.
Diagonals give : A000108 A001700 A002457 A002802 A002803, A000012 A000217 A034827 A000910 A088625 A088626.
Sequence in context: A030103 A105640 A090299 * A172381 A089302 A049020
Adjacent sequences: A060690 A060691 A060692 * A060694 A060695 A060696
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Frederic Chapoton (chapoton(AT)math.jussieu.fr), Apr 20 2001
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 21 2001
New description from Philippe Deleham, Aug 12, 2003
|
| |
|
|
