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Search: id:A114489
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| A114489 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n that have k valleys at level 1. |
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+0
1
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| 1, 1, 2, 4, 1, 9, 4, 1, 22, 14, 5, 1, 58, 46, 21, 6, 1, 163, 149, 80, 29, 7, 1, 483, 484, 292, 124, 38, 8, 1, 1494, 1589, 1044, 498, 179, 48, 9, 1, 4783, 5288, 3701, 1928, 780, 246, 59, 10, 1, 15740, 17848, 13096, 7304, 3237, 1152, 326, 71, 11, 1, 52956, 61060, 46428
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Rows 0 and 1 contain one term each; row n contains n-1 terms (n>=2). Row sums are the Catalan numbers (A000108). Column 0 yields A059019. Sum(k*T(n,k),k=0..n-1) = 6binomial(2n-1,n-3)/(n+3) (A003517).
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FORMULA
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G.f.=(1-tzC)/[(1-z)(1-tzC)-z^2*C], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(4,1)=4 because we have UU(DU)DDUD, UDUU(DU)DD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the valleys at level 1 are shown between parentheses.
Triangle starts:
1;
1;
2;
4,1;
9,4,1;
22,14,5,1;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=(1-t*z*C)/(1-t*z*C-z+t*z^2*C-z^2*C): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: 1; 1; for n from 2 to 12 do seq(coeff(t*P[n], t^j), j=1..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A059019, A003517.
Sequence in context: A057551 A019823 A092107 this_sequence A101974 A097607 A132893
Adjacent sequences: A114486 A114487 A114488 this_sequence A114490 A114491 A114492
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 01 2005
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