%I A061554
%S A061554 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,
%T A061554 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,
%U A061554 210,120,120,45,45,10,10,1,1,462,462,330,330,165,165,55,55,11,11,1,1
%N A061554 Square table read by antidiagonals: a(n,k)=C(n+k,[k/2]).
%C A061554 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
%C A061554 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
%C A061554 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
%C A061554 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
%C A061554 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
%C A061554 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 A061554 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]
%C A061554 Triangle read by rows = partial sums of A053121 terms starting fromt 
               the right. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 24 2008]
%F A061554 As a triangle: T(n,k) = C(n,m) where m = floor[(n+1)/2 - (-1)^(n-k)*(k+1)/
               2].
%F A061554 a(0, k)=C(k, [k/2])=A001405(k); for n>0 T(n, k)=T(n+1, k-2)+T(n-1, k).
%F A061554 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
%F A061554 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
%F A061554 Sum{k, 0<=k<=n}T(n,k)=2^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 27 2007
%e A061554 Triangle version begins:
%e A061554 1;
%e A061554 1, 1;
%e A061554 2, 1, 1;
%e A061554 3, 3, 1, 1;
%e A061554 6, 4, 4, 1, 1;
%e A061554 10, 10, 5, 5, 1, 1;
%e A061554 20, 15, 15, 6, 6, 1, 1;
%e A061554 35, 35, 21, 21, 7, 7, 1, 1;
%e A061554 70, 56, 56, 28, 28, 8, 8, 1, 1;
%e A061554 126, 126, 84, 84, 36, 36, 9, 9, 1, 1;
%e A061554 252, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1;
%e A061554 462, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1; ...
%e A061554 Matrix inverse begins:
%e A061554 1;
%e A061554 -1, 1;
%e A061554 -1, -1, 1;
%e A061554 1, -2, -1, 1;
%e A061554 1, 2, -3, -1, 1;
%e A061554 -1, 3, 3, -4, -1, 1;
%e A061554 -1, -3, 6, 4, -5, -1, 1;
%e A061554 1, -4, -6, 10, 5, -6, -1, 1;
%e A061554 1, 4, -10, -10, 15, 6, -7, -1, 1; ...
%e A061554 Contribution from Paul Barry (pbarry(AT)wit.ie), May 21 2009: (Start)
%e A061554 Production matrix is
%e A061554 .1, 1,
%e A061554 .1, 0, 1,
%e A061554 .0, 1, 0, 1,
%e A061554 .0, 0, 1, 0, 1,
%e A061554 .0, 0, 0, 1, 0, 1,
%e A061554 .0, 0, 0, 0, 1, 0, 1,
%e A061554 .0, 0, 0, 0, 0, 1, 0, 1 (End)
%o A061554 (PARI) T(n,k)=binomial(n,(n+1)\2-(-1)^(n-k)*((k+1)\2))
%Y A061554 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.
%Y A061554 Cf. A007318, A107430, A125094, A037952, A066170.
%Y A061554 Sequence in context: A128545 A034364 A090011 this_sequence A088326 A124975 
               A129439
%Y A061554 Adjacent sequences: A061551 A061552 A061553 this_sequence A061555 A061556 
               A061557
%K A061554 nonn,tabl
%O A061554 0,4
%A A061554 Henry Bottomley (se16(AT)btinternet.com), May 17 2001
%E A061554 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006