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A097610
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Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.
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6
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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, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1, 0, 126, 0, 420, 0, 252, 0, 36, 0, 1, 42, 0, 630, 0, 1050, 0, 420, 0, 45, 0, 1, 0, 462, 0, 2310, 0, 2310, 0, 660, 0, 55, 0, 1, 132, 0, 2772, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are the Motzkin 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.
Exponential Riordan array [Bessel_I(1,2x)/x,x]. [From Paul Barry (pbarry(AT)wit.ie), Mar 24 2010]
Diagonal sums are the aerated large Schroeder numbers. [From Paul Barry (pbarry(AT)wit.ie), Apr 21 2010]
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REFERENCES
| G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169.
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FORMULA
| 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
G.f.: 1/(1-x*y-x^2/(1-x*y-x^2/(1-x*y-x^2/..... (continued fraction); [From Paul Barry (pbarry(AT)wit.ie), Dec 15 2008]
Sum_{k, 0<=k<=n}T(n,k)*4^k = A005572(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 03 2009]
T(n,k)= A007318(n,k)*A126120(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 12 2009]
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EXAMPLE
| 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).
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MAPLE
| 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);
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CROSSREFS
| Cf. A001006, A000108. A124027 is an essentially identical triangle.
Sequence in context: A021336 A100749 A124027 * A161556 A129555 A147755
Adjacent sequences: A097607 A097608 A097609 * A097611 A097612 A097613
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 30 2004
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