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) -> A039599; (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

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, December 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

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

Adjacent sequences: A039596 A039597 A039598 this_sequence A039600 A039601 A039602

Sequence in context: A065883 A071975 A055905 this_sequence A011357 A080409 A030103

nonn,tabl,easy,nice

njas

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