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

A053121^2 = triangle A145973. Convolved with A001405 = triangle A153585. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 13 2009: (Start)

By columns without the zeros, n-th row = A000108 convolved with itself

n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then

n-th row = coefficients of A^(n+1). (End)

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

Sum_{k, 0<=k<=n} T(n,k)^x = A000027(n+1), A001405(n), A000108(n), A003161(n), A129123(n) for x = 0,1,2,3,4 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009]

Sum_{k, 0<=k<=n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -1, 0, 1, 2, 3 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]

.......|...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.

Contribution from Paul Barry (pbarry(AT)wit.ie), May 29 2009: (Start)

Production matrix is

0, 1,

1, 0, 1,

0, 1, 0, 1,

0, 0, 1, 0, 1,

0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 0, 1, 0, 1,

0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)

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.

A145973, A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]

Sequence in context: A125921 A029299 A049803 this_sequence A113408 A022337 A025687

Cf. A108786 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009]

Adjacent sequences: A053118 A053119 A053120 this_sequence A053122 A053123 A053124

easy,nice,tabl,nonn,new

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