%I A053121
%S A053121 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,5,0,4,0,1,5,0,9,0,5,0,1,0,14,0,14,0,6,
%T A053121 0,1,14,0,28,0,20,0,7,0,1,0,42,0,48,0,27,0,8,0,1,42,0,90,0,75,0,35,0,9,
%U A053121 0,1,0,132,0,165,0,110,0,44,0,10,0,1,132,0,297,0,275,0,154,0,54,0,11,0
%N A053121 Catalan triangle (with 0's). Inverse lower triangular matrix of A049310(n,
               m) (coefficients of Chebyshev's S polynomials).
%C A053121 "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]
%C A053121 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).
%C A053121 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)
%C A053121 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.
%C A053121 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.
%C A053121 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
%C A053121 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
%C A053121 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
%C A053121 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
%C A053121 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
%C A053121 A053121^2 = triangle A145973. Convolved with A001405 = triangle A153585. 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]
%C A053121 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 13 2009: 
               (Start)
%C A053121 By columns without the zeros, n-th row = A000108 convolved with itself
%C A053121 n times; equivalent to A = (1 + x + 2x^2 + 5x^3 + 14x^4 + ...), then
%C A053121 n-th row = coefficients of A^(n+1). (End)
%D A053121 I. Bajunaid et al., Function series, Catalan numbers and random walks 
               on trees, Amer. Math. Monthly 112 (2005), 765-785.
%D A053121 J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, 
               Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
%D A053121 E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, 
               European Journal of Combinatorics 28 (2007), 481-498 (see pp. 486 
               and 498).
%D A053121 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.
%D A053121 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.
%D A053121 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans 
               in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%D A053121 L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, 
               Discrete Appl. Maths. 34 (1991) 229-239.
%D A053121 W.-J. Woan, Area of Catalan Paths, Discrete Math., 226 (2001), 439-444.
%H A053121 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/
               Papers/infjumps.ps">Lattice paths with an infinite set of jumps</
               a>
%H A053121 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http:/
               /www.math.wvu.edu/~mays/Papers/apd2.ps">A Pascal rhombus</a>, Fibonacci 
               Quarterly, 35 (1997), 318-328.
%H A053121 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A053121 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);
%F A053121 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.
%F A053121 G.f. for m-th column: c(x^2)*(x*c(x^2))^m, where c(x) = g.f. for Catalan 
               numbers A000108.
%F A053121 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
%F A053121 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
%F A053121 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
%F A053121 Sum_{k, 0<=k<=n}T(n,k)*(k+1)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 22 2007
%F A053121 Sum_{j, j>=0}T(n,j)*binomial(j,k)= A054336(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 30 2007
%F A053121 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
%F A053121 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]
%e A053121 .......|...1
%e A053121 .......|.......1
%e A053121 .......|...1.......1
%e A053121 .......|.......2.......1
%e A053121 .......|...2.......3.......1
%e A053121 .......|.......5.......4.......1
%e A053121 .......|...5.......9.......5.......1
%e A053121 .......|......14......14.......6.......1
%e A053121 .......|..14......28......20.......7.......1
%e A053121 .......|......42......48......27.......8.......1
%e A053121 {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.
%e A053121 Contribution from Paul Barry (pbarry(AT)wit.ie), May 29 2009: (Start)
%e A053121 Production matrix is
%e A053121 0, 1,
%e A053121 1, 0, 1,
%e A053121 0, 1, 0, 1,
%e A053121 0, 0, 1, 0, 1,
%e A053121 0, 0, 0, 1, 0, 1,
%e A053121 0, 0, 0, 0, 1, 0, 1,
%e A053121 0, 0, 0, 0, 0, 1, 0, 1,
%e A053121 0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A053121 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
%p A053121 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
%Y A053121 Cf. A008315, A049310, A033184, A000108, A001405. Another version: A008313.
%Y A053121 Variant without zero-diagonals: A033184 and with rows reversed: A009766.
%Y A053121 A145973, A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 
               2008]
%Y A053121 Sequence in context: A125921 A029299 A049803 this_sequence A113408 A022337 
               A025687
%Y A053121 Cf. A108786 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009]
%Y A053121 Adjacent sequences: A053118 A053119 A053120 this_sequence A053122 A053123 
               A053124
%K A053121 easy,nice,tabl,nonn,new
%O A053121 0,8
%A A053121 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)