0,8

Row sums are the Motkin numbers (A001006). Column 0 gives the aerated Catalan numbers (A000108).

Applied as a matrix to the power series r^n, it gives the sequence which counts r-colored Motzkin paths of length n; equivalently, sum{k=0..n,C(n,k)C((n-k)/2)(1+(-1)^(n-k)r^k/2}=sum{k=0..floor(k/2),C(n,2k)C(k)r^(n-2k)}. - Paul Barry (pbarry(AT)wit.ie), May 18 2005

Let P_n(x)=Sum_{k, 0<=k<=n}T(n,k)*x^k .P_0(x)=1, P_1(x)=x, P_n(x)=x*P_(n-1)(x)+Sum_{j, 0<=j<=(n-2)}P_j(x)*P_(n-2-j)(x); essentially the same as A124027 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 03 2007

Comment from Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 31 2006: G. J. Chaitin's numbers of s-expressions of size n are given by the coefficients of polynomials p(k, x) satisfying : p(k, x) = Sum[p(j, x)*p(k - j, x), {j, 2, k - 1}]. The coefficients of these polynomials also give (essentially) the triangle shown here.

G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169.

G:=[1-tz-sqrt(1-2tz+t^2*z^2-4z^2)]/(2z^2). T(n, k)=n!/[k!((n-k)/2)!((n-k)/2-1)! ] = A055151(n, (n-k)/2) if n-k is a nonnegative even number; otherwise T(n, k)=0.

T(n, k)=if(k<=n, C(n, k)C((n-k)/2)(1+(-1)^(n-k))/2, 0) - Paul Barry (pbarry(AT)wit.ie), May 18 2005

T(n,k)=A121448(n,k)/2^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2006

Sum_[k,0<=k<=n}T(n,k)*2^k = A000108(n+1) - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 22 2006

Sum_{k, 0<=k<=n}T(n,k)*3^k = A002212(n+1) - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 03 2007

Triangle begins:

1;

0,1;

1,0,1;

0,3,0,1;

2,0,6,0,1;

0,10,0,10,0,1;

5,0,30,0,15,0,1;

Row n has n+1 terms.

T(4,2)=6 because we have HHUD, HUDH, UDHH, HUHD, UHDH, UHHD, where U=(1,1),

D=(1,-1) and H=(1,0).

G:=(1-t*z-sqrt(1-2*t*z+t^2*z^2-4*z^2))/2/z^2: Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..13); Maple program for the triangular array: T:=proc(n, k) if n-k mod 2 = 0 and k<=n then n!/k!/((n-k)/2)!/((n-k)/2+1)! else 0 fi end: TT:=(n, k)->T(n-1, k-1): matrix(10, 10, TT);

Cf. A001006, A000108. A124027 is an essentially identical triangle.

Sequence in context: A021336 A100749 A124027 this_sequence A129555 A136748 A049765

Adjacent sequences: A097607 A097608 A097609 this_sequence A097611 A097612 A097613

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 30 2004