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Search: id:A109189
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| A109189 |
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Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps at level zero. (A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).). |
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+0
2
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| 1, 0, 1, 2, 0, 1, 2, 4, 0, 1, 8, 4, 6, 0, 1, 16, 20, 6, 8, 0, 1, 46, 40, 36, 8, 10, 0, 1, 114, 128, 72, 56, 10, 12, 0, 1, 310, 324, 254, 112, 80, 12, 14, 0, 1, 822, 932, 654, 432, 160, 108, 14, 16, 0, 1, 2238, 2540, 1986, 1128, 670, 216, 140, 16, 18, 0, 1, 6094, 7164, 5546
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OFFSET
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0,4
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COMMENT
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Row sums yield the central trinomial coefficients (A002426). T(n,0)=A109190(n). sum(k*T(n,k),k=0..n)=A015518(n).
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FORMULA
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G.f.= 1/(1-tz-2z^2*M), where M=1+zM+z^2*M^2=[1-z-sqrt(1-2z-3z^2)]/(2z^2) is the g.f. of the Motzkin numbers (A001006).
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EXAMPLE
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T(4,1)=4 because we have (h)uhd, (h)dhu, uhd(h) and dhu(h), where u=(1,1),d=(1,-1), h=(1,0) and the (1,0) steps at level 0 are shown between parentheses.
Triangle begins:
1;
0,1;
2,0,1;
2,4,0,1;
8,4,6,0,1;
16,20,6,8,0,1;
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MAPLE
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M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-t*z-2*z^2*M): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od;
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CROSSREFS
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Cf. A002426, A109190, A015518, A001006.
Sequence in context: A115346 A140531 A117316 this_sequence A144172 A166692 A046766
Adjacent sequences: A109186 A109187 A109188 this_sequence A109190 A109191 A109192
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2005
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