0,2

T(n,k)=number of leaves at level k+1 in all ordered trees with n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2005

Riordan array ((1-2x-sqrt(1-4x))/(2x^2),(1-2x-sqrt(1-4x))/(2x)). Inverse array is A053122. - Paul Barry (pbarry(AT)wit.ie), Mar 17 2005

T(n,k)=number of walks of n steps, each in direction N, S,W,or E, starting at the origin, remaining in the upper half-plane and ending at height k (see the R. K. Guy reference, p. 5). Example: T(3,2)=6 because we have ENN, WNN, NEN, NWN, NNE and NNW. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 15 2005

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)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

Number of 2n+1 step walks from (0,0) to (2n+1,2k+1) and consisting of step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant . Example : T(2,0)=5 because we have uuudd, uudud, uuddu, uduud, ududu ; T(2,1)=4 because we have uuuud, uuudu, uuduu, uduuu ; T(2,2)=1 because we have uuuuu . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 18 2007

Triangle read by rows:T(n,k)=number of lattice paths from (0,0) to (n,k)that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1)and two types of steps H=(1,0); example: T(3,1)=14 because we have UDU, UUD, 4 HHU paths, 4 HUH paths and 4 UHH paths . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 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

With offset [1,1] this is the (ordinary) convolution triangle a(n,m) with o.g.f. of column nr. m given by (c(x)-1)^m, where c(x) is the o.g.f. for Catalan numbers A000108. See the Riordan comment by P. Barry.

T(n, k) is also the number of order-preserving full transformations (of an n-chain) with exactly k fixed points. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

T(n,k)/2^(2n+1)=coefficients of the maximally flat lowpass digital differentiator of the order N=2n+3. [From Pavel Holoborodko (pavel(AT)holoborodko.com), Dec 19 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

Higgins, Peter M. Combinatorial results for semigroups of order-preserving mappings. Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

Row n: C(2n, n-k)-C(2n, n-k-2).

a(n, k) = C(2n+1, n-k)*2*(k+1)/(n+k+2) = A050166(n, n-k) = a(n-1, k-1)+2*a(n-1, k)+a(n-1, k+1) [with a(0, 0) = 1 and a(n, k) = 0 if n<0 or n<k]. - Henry Bottomley (se16(AT)btinternet.com), Sep 24 2001

T(n, 0) = A000108(n+1), T(n, k) = 0 if n<k; for k>0, T(n, k) = Sum_{j=1..n} T(n-j, k-1)*A000108(j) . G.f. for column k : Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+2) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108 . Sum_{k>=0} T(m, k)*T(n, k) = A000108(m+n+1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004

T(n, k) = A009766(n+k+1, n-k) = A033184(n+k+2, 2k+2) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004

Sum_{j>=0} T(k, j)*A039599(n-k, j) = A028364(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 04 2004

Antidiagonal sum_{k=0..n} T(n-k, k) = A000957(n+3). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jun 05 2005

The triangle may also be generated from M^n * [1,0,0,0...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [2,2,2...] in the main diagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2006

G.f.=G(t,x)=C^2/(1-txC^2), where C=[1-sqrt(1-4x)]/(2x) is the Catalan function. From here G(-1,x)=C, i.e. the alternating row sums are the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 20 2007

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000108(n+1), A001700(n), A049027(n+1), A076025(n+1), A076026(n+1) for x=0,1,2,3,4 respectively (see square array in A067345) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 21 2007

Sum_{k, 0<=k<=n}T(n,k)*(k+1)=4^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

Sum_{j, j>=0}T(n,j)*binomial(j,k)=A035324(n,k), A035324 with offset 0 (0<=k<=n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

T(n,k)=A053121(2*n+1,2*k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 18 2007

T(n,k) = A039599(n,k)+A039599(n,k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 11 2007

Sum_{k, 0<=k<=n+1}T(n+1,k)*k^2 = A029760(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 16 2007

Sum_{k, 0<=k<=n}T(n,k)*A059841(k)=A000984(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]

G.f.: 1/(1-xy-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction).

Triangle starts:

1;

2,1;

5,4,1;

14,14,6,1;

42,48,27,8,1;

Cf. A008313, A039599.

Row sums : A001700

Sequence in context: A054456 A096164 A104710 this_sequence A128738 A126181 A154930

Adjacent sequences: A039595 A039596 A039597 this_sequence A039599 A039600 A039601

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Clark Kimberling (ck6(AT)evansville.edu)

Typo in one entry corrected by Philippe DELEHAM, Dec 16 2007