0,4

T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E = (1,0) and N = (0,1) which touch but do not cross the line x - y = k and only situated above this line; example : T(3,2) = 5 because we have EENNNE, EENNEN, EENENN, ENEENN, NEEENN. - Philippe DELEHAM, May 23 2005

The matrix inverse of this triangle is the triangular matrix T(n,k) = (-1)^(n+k)* A085478(n,k). - Philippe DELEHAM, May 26 2005

Essentially the same as A050155 except with a leading diagonal A000108 (Catalan numbers) 1, 1, 2, 5, 14, 42, 132, 429, . . . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005

Number of Grand Dyck paths of semilength n and having k downward returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). Example: T(3,2)=5 because we have u(d)uud(d),uud(d)u(d),u(d)u(d)du,u(d)duu(d) and duu(d)u(d) (the downward returns to the x-axis are shown between parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 06 2006

Riordan array (c(x),x*c(x)^2) where c(x) is the g.f. of A000108 ; inverse array is (1/(1+x),x/(1+x)^2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 12 2007

The triangle may also be generated from M^n*[1,0,0,0,0,0,0,0,...], where M is the infinite tridiagonal matrix with all 1's in the super and subdiagonals and [1,2,2,2,2,2,2,...] in the main diagonal . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26 2007

Inverse binomial matrix applied to A124733 . Binomial matrix applied to A089942 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26 2007

Number of standard tableaux of shape (n+k,n-k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Comment from Phipppe DELEHAM, 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.

The table U(n,k)=Sum_{j, 0<=j<=n}T(n,j)*k^j is given in A098474 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 29 2007

Sequence read mod 2 gives A127872 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 12 2007

Number of 2n step walks from (0,0) to (2n,2k)and consisting of step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant . Example :T(3,0)=5 because we have uuuddd, uududd, ududud, uduudd, uuddud ; T(3,1)=9 because we have uuuudd, uuuddu, uuudud, ududuu, uuduud, uduudu, uudduu, uduuud, uududu ; T(3,2)=5 because we have uuuuud, uuuudu, uuuduu, uuduuu, uduuuu ; T(3,3)=1 because we have uuuuuu . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 17 2007, Apr 18 2007

Triangular matrix, read by rows, equal to the matrix inverse of triangle A129818 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2007

Let Sum_{n>=0}a(n)*x^n = (1+x)/(1-mx+x^2)= o.g.f. of A_m, then Sum_{k, 0<=k<=n}T(n,k)*a(k)= (m+2)^n. Related expansions of A_m are : A099493, A033999, A057078, A057077, A057079, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, A097783, A077416, A126866, A028230, A161591, for m= -3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2009]

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 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.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

T. D. Noe, Rows n=0..50 of triangle, flattened

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].

Row n: C(2n-1, n-k)-C(2n-1, n-k-2).

Triangle T(n, k) read by rows; given by A000012 DELTA A000007, where DELTA is Deleham's operator defined in A084938.

T(n, k) = C(2*n, n-k)*(2*k+1)/(n+k+1). Sum(k>=0; T(n, k)*T(m, k) = A000108(n+m)); A000108: numbers of Catalan. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 22 2003

T(n, 0) = A000108(n); T(n, k) = 0 if k>n; for k>0, T(n, k) = Sum_{j=1..n) T(n-j, k-1)*A000108(j). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 03 2004

T(n, k) = A009766(n+k, n-k) = A033184(n+k+1, 2k+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 03 2004

G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+1) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 03 2004

T(0, 0) = 1, T(n, k) = 0 if n<0 or n<k; T(n, 0) = T(n-1, 0) + T(n-1, 1); for k>=1, T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004

a(n) + a(n+1) = 1 + A000108(m+1) if n = m*(m+3)/2; a(n) + a(n+1) = A039598(n) otherwise. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004

T(n, k) = A050165(n, n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 27 2004

Sum_{j>=0} T(n-k, j)*A039598(k, j) = A028364(n, k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 04 2004

Matrix inverse of the triangle T(n, k) = (-1)^(n+k)*binomial(n+k, 2*k) = (-1)^(n+k)*A085478(n, k). - Philippe Deleham (kolotoko(AT)wanadoo.fr)

Sum_{k, 0<=k<=n} T(n, k)*x^k = A000108(n), A000984(n), A007854(n), A076035(n), A076036(n) for x = 0, 1, 2, 3, 4 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 19 2005

Sum_{k, 0<=k<=n} (2*k+1)*T(n, k) = 4^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 22 2005

T(n, k)*(-2)^(n-k) = A114193(n, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2005

Sum_{k>=h}T(n,k)=binomial(2n,n-h). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 30 2006

T(n,k)=(2k+1)*binomial(2n,n-k)/(n+k+1). G.f.=G(t,z)=1/[1-(1+t)zC], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 06 2006

Sum_{k, 0<=k<=n} T(n,k)*5^k = A127628(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 22 2007

Sum_{k, 0<=k<=n} T(n,k)*7^k = A115970(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 26 2007

T(n,k)=Sum_{j, 0<=j<=n-k}A106566(n+k,2*k+j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 12 2007

Sum_{k, 0<=k<=n}T(n,k)*6^k = A126694(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 16 2007

Sum_{k, 0<=k<=n}T(n,k)*A000108(k)=A007852(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{k, 0<=k<=[n/2]}T(n-k,k)=A000958(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{k, 0<=k<=n}T(n,k)*(-1)^k=A000007(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{k, 0<=k<=n}T(n,k)*(-2)^k = (-1)^n*A064310(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

T(2n,n)=A126596(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{k, 0<=k<=n}T(n,k)*(-x)^k=A000007(n),A126983(n),A126984(n),A126982(n),A126986(n),A126987(n),A127017(n),A127016(n),A126985(n),A127053(n) for x=1,2,3,4,5,6,7,8,9,10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007

Sum_{j, j>=0}T(n,j)*binomial(j,k)= A116395(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

T(n,k)=Sum_{j, j>=0}A106566(n,j)*binomial(j,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007

T(n,k)=Sum_{j, j>=0}A127543(n,j)*A038207(j,k}. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 03 2007

Sum_{k, 0<=k<=[n/2]}T(n-k,k)*A000108(k)=A101490(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 12 2007

T(n,k)=A053121(2*n,2*k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 17 2007, Apr 18 2007

Sum_{k, 0<=k<=n}T(n,k)*sin((2*k+1)*x)=sin(x)*(2*cos(x))^(2*n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 17 2007, Apr 18 2007

T(n,n-k)= Sum_{j, j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 05 2007

Sum_{j, j>=0}A110506(n,j)*binomial(j,k)=Sum_{j, j>=0}A110510(n,j)*A038207(j,k)=T(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 25 2007

Sum_{j, j>=0}A110518(n,j)*A027465(j,k)=Sum_{j, j>=0}A110519(n,j)*A038207(j,k)=T(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 25 2007

Sum_{k, 0<=k<=n}T(n,k)*A001045(k)=A049027(n), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007

Sum_{k, 0<=k<=n}T(n,k)*a(k)=(m+2)^n if Sum_{k, k>=0}a(k)*x^k = (1+x)/(x^2-m*x+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007

Sum_{k, 0<=k<=n}T(n,k)*A040000(k) = A001700(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007

Sum_{k, 0<=k<=n}T(n,k)*A122553(k) = A051924(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007

Sum_{k, 0<=k<=n}T(n,k)*A123932(k) = A051944(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007

Sum_{k, 0<=k<=n}T(n,k)*k^2 = A000531(n), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007

Sum_{k, 0<=k<=n}T(n,k)*A000217(k)=A002457(n-1), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007

Sum{j, j>=0}binomial(n,j)*T(j,k)= A124733(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 16 2007

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007

Sum_{k, 0<=k<=n}T(n,k)*A005043(k)=A127632(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2007

Sum_{k, 0<=k<=n}T(n,k)*A132262(k)=A089022(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2007

T(n,k)+T(n,k+1)=A039598(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2007

T(n,k) = A128899(n,k)+A128899(n,k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2007

Sum_{k, 0<=k<=n}T(n,k)*A015518(k) = A076025(n), for n>=1. Also Sum_{k, 0<=k<=n}T(n,k)*A015521(k) = A076026(n), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2007

Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*x^(n-k) = A033999(n), A000007(n), A064062(n), A110520(n), A132863(n), A132864(n), A132865(n), A132866(n), A132867(n), A132869(n), A132897(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2007

Sum_{k, 0<=k<=n}T(n,k)*(-1)^(k+1)*A000045(k)=A109262(n), A000045:= Fibonacci numbers. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

Sum_{k, 0<=k<=n}T(n,k)*A000035(k)*A016116(k)=A143464(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

Sum_{k, 0<=k<=n}T(n,k)*A016116(k)=A101850(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2008]

Sum_{k, 0<=k<=n}T(n,k)*A010684(k)=A100320(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2008]

Sum_{k, 0<=k<=n}T(n,k)*A000034(k)=A029651(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2008]

Sum_{k, 0<=k<=n}T(n,k)*A010686(k)=A144706(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

Sum_{k, 0<=k<=n}T(n,k)*A006130(k-1)=A143646(n), with A006130(-1)=0 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 01 2008]

T(n,2*k)+T(n,2*k+1)=A118919(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2008]

Sum_{k, 0<=k<=j}T(n,k)=A050157(n,j). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]

Sum_{k, 0<=k<=2}T(n,k) = A026012(n); Sum_{k, 0<=k<=3}T(n,k)=A026029(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]

Sum_{k, 0<=k<=n}T(n,k)*A000045(k+2)=A026671(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*A000045(k+1)=A026726(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*A057078(k)=A000012(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 27 2009]

Sum_{k, 0<=k<=n}T(n,k)*A108411(k)=A155084(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*A057077(k)= 2^n = A000079(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*A057079(k) = 3^n = A000244(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 11 2009]

Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*A123344(k)= A000957(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2009]

Triangle begins:

1;

1, 1;

2, 3, 1;

5, 9, 5, 1;

14, 28, 20, 7, 1;

42, 90, 75, 35, 9, 1;

T:=(n, k)->(2*k+1)*binomial(2*n, n-k)/(n+k+1): for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 06 2006

Diagonals give : A000108 A000245 A000344 A000588 A001392 A000589 A000590, A000012 A005408 A014107(n>1) Row sums : A000984

Cf. A008313 A039598 A084938 A000007

Sequence in context: A055905 A147703 A147747 this_sequence A154380 A155083 A011357

Adjacent sequences: A039596 A039597 A039598 this_sequence A039600 A039601 A039602

nonn,tabl,easy,nice,new

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

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