0,4

Equivalently, a triangle read by rows, where the rows are obtained by sorting the elements of the rows of Pascal's triangle (A007318) into descending order. - Philippe DELEHAM, May 21 2005

Equivalently, as a triangle read by rows, this is T(n,k)=binomial(n,floor((n-k)/2)); column k then has e.g.f. Bessel_I(k,2x)+Bessel_I(k+1,2x). - Paul Barry (pbarry(AT)wit.ie), Feb 28 2006

Antidiagonal sums are A037952(n+1) = C(n+1,[n/2]). Matrix inverse is the row reversal of triangle A066170. Eigensequence is A125094(n) = Sum_{k=0..n-1} A125093(n-1,k)*A125094(k). - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 20 2006

Riordan array (1/(1-x-x^2*c(x^2)),x*c(x^2)); where c(x)=g.f.for Catalan numbers A000108 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 17 2007

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-1,0)+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 27 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

T(n,k) is the number of paths from (0,k) to some (n,m) which never dip below y=0, touch y=0 at least once and are made up only of the steps (1,1) and (1,-1). This can be proved using the recurrence supplied by DELEHAM. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 15 2008]

Triangle read by rows = partial sums of A053121 terms starting fromt the right. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 24 2008]

As a triangle: T(n,k) = C(n,m) where m = floor[(n+1)/2 - (-1)^(n-k)*(k+1)/2].

a(0, k)=C(k, [k/2])=A001405(k); for n>0 T(n, k)=T(n+1, k-2)+T(n-1, k).

n-th row = M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super and subdiagonals and (1,0,0,0...) in the main diagonal. V = the infinite vector [1,0,0,0...]. Example: (3,3,1,1,0,0,0...) = M^3 * V. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 04 2006

Sum_{k, 0<=k<=n}T(m,k)*T(n,k)=T(m+n,0)=A001405(m+n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26 2007

Sum{k, 0<=k<=n}T(n,k)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007

Triangle version begins:

1;

1, 1;

2, 1, 1;

3, 3, 1, 1;

6, 4, 4, 1, 1;

10, 10, 5, 5, 1, 1;

20, 15, 15, 6, 6, 1, 1;

35, 35, 21, 21, 7, 7, 1, 1;

70, 56, 56, 28, 28, 8, 8, 1, 1;

126, 126, 84, 84, 36, 36, 9, 9, 1, 1;

252, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1;

462, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1; ...

Matrix inverse begins:

1;

-1, 1;

-1, -1, 1;

1, -2, -1, 1;

1, 2, -3, -1, 1;

-1, 3, 3, -4, -1, 1;

-1, -3, 6, 4, -5, -1, 1;

1, -4, -6, 10, 5, -6, -1, 1;

1, 4, -10, -10, 15, 6, -7, -1, 1; ...

Contribution from Paul Barry (pbarry(AT)wit.ie), May 21 2009: (Start)

Production matrix is

.1, 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 (End)

(PARI) T(n, k)=binomial(n, (n+1)\2-(-1)^(n-k)*((k+1)\2))

Rows are A001405, A037952, A037955, A037951, A037956, A037953, A037957 etc. Columns are truncated pairs of A000012, A000027, A000217, A000292, A000332, A000389, A000579, etc. Main diagonal is alternate values of A051036.

Cf. A007318, A107430, A125094, A037952, A066170.

Sequence in context: A128545 A034364 A090011 this_sequence A088326 A124975 A129439

Adjacent sequences: A061551 A061552 A061553 this_sequence A061555 A061556 A061557

nonn,tabl

Henry Bottomley (se16(AT)btinternet.com), May 17 2001

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006