0,8

"The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]

G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n): c(z^2)/(1-x*z*c(z^2)). Row sums (x=1): A001405 (central binomial).

In the language of the Shapiro et al. reference such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. Ginv(x) of the m=0 column of the inverse of a given Bell-matrix (here A049310)

is obtained from its g.f. of the m=0 column (here G(x)=1/(1+x^2)) by Ginv(x)=(f^{(-1)}(x))/x, with f(x) := x*G(x) and f^{(-1)}is the compositional inverse function of f (here one finds, with Ginv(0)=1, c(x^2)). See the Shapiro et al. reference.

Walks with a wall: triangle of number of n step walks from (0,0) to (n,m) where each step goes from (a,b) to (a+1,b+1) or (a+1,b-1) and the path stays in the nonnegative quadrant.

Row sums of squares equals the Catalan sequence (A000108); for row 6: A000108(6) = 5^2 + 0^2 + 9^2 + 0^2 + 5^2 + 0^2 + 1^2 = 132. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 23 2005

Number of involutions of {1,2,...,n} that avoid the patterns 132 and have exactly k fixed points. Example: T(4,2)=3 because we have 2134, 4231, and 3214. Number of involutions of {1,2,...,n} that avoid the patterns 321 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1324, and 2134. Number of involutions of {1,2,...,n} that avoid the patterns 213 and have exactly k fixed points. Example: T(4,2)=3 because we have 1243, 1432, and 4231. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006

Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007

Riordan array (c(x^2),xc(x^2)), where c(x) is the g.f. of Catalan numbers A000108 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 25 2007

I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.

J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)

E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 and 498).

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and Related Sequences Arising from Inverses of Pascal's Triangle Matrices, Fibonacci Quart. 14 (1976) 395-405.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fibonacci Quart. 38,5 (2000) 408-419; Note 4, pp. 414-415.

A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

Index entries for sequences related to Chebyshev polynomials.

a(n, m) := 0 if n<m or n-m odd, else a(n, m) = (m+1)*binomial(n+1, (n-m)/2)/(n+1);

a(n, m) = (4*(n-1)*a(n-2, m) + 2*(m+1)*a(n-1, m-1))/(n+m+2), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1), a(1, 0)=0.

G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan numbers A000108.

a(n, m)=a(n-1, m-1)+a(n-1, m+1) if n>0 and m >= 0, a(0, 0)=1, a(0, m)=0 if m>0, a(n, m)=0 if m<0 - Henry Bottomley (se16(AT)btinternet.com), Jan 25 2001

Sum_{k>=0} T(m, k)*T(n, k) = 0 if m+n is odd; Sum_{k>=0} T(m, k)*T(n, k) = A000108((m+n)/2) if m+n is even . - Philippe DELEHAM, May 26 2005

T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i, C(i,j)*(C(i-j,j+k)-C(i-j,j+k+2))}}; Column k has e.g.f. BesselI(k,2x)-BesselI(k+2,2x); - Paul Barry (pbarry(AT)wit.ie), Feb 16 2006

Sum_{k, 0<=k<=n}T(n,k)*(k+1)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{j, j>=0}T(n,j)*binomial(j,k)= A054336(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

T(2*n+1,2*k+1)=A039598(n,k), T(2*n,2*k)=A039599(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007

.......|...1

.......|.......1

.......|...1.......1

.......|.......2.......1

.......|...2.......3.......1

.......|.......5.......4.......1

.......|...5.......9.......5.......1

.......|......14......14.......6.......1

.......|..14......28......20.......7.......1

.......|......42......48......27.......8.......1

{1}; {0,1}; {1,0,1}; {0,2,0,1}; {2,0,3,0,1};... E.g. fourth row corresponds to polynomial p(3,x)= 2*x+x^3.

T:=proc(n, k): if n+k mod 2 = 0 then (k+1)*binomial(n+1, (n-k)/2)/(n+1) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2006

Cf. A008315, A049310, A033184, A000108, A001405. Another version: A008313.

Variant without zero-diagonals: A033184 and with rows reversed: A009766.

Adjacent sequences: A053118 A053119 A053120 this_sequence A053122 A053123 A053124

Sequence in context: A125921 A029299 A049803 this_sequence A113408 A022337 A025687

easy,nice,tabl,nonn

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)