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Search: id:A114462
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| A114462 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)). |
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5
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| 1, 1, 1, 1, 2, 3, 6, 7, 1, 18, 19, 5, 54, 59, 18, 1, 166, 191, 65, 7, 522, 631, 242, 34, 1, 1670, 2123, 906, 154, 9, 5418, 7247, 3395, 680, 55, 1, 17786, 25011, 12746, 2932, 300, 11, 58974, 87071, 47931, 12414, 1540, 81, 1, 197226, 305275, 180439, 51878, 7552
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OFFSET
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0,5
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k),k=0..floor(n/2))=binomial(2n-3,n-1)-binomial(2n-4,n)=A077587(n-2) (n>=2). Column 0 yields A114464.
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FORMULA
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G.f. G=G(t, z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.
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EXAMPLE
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T(4,1)=7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,
UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).
Triangle starts:
1;
1;
1,1;
2,3;
6,7,1;
18,19,5;
54,59,18,1;
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MAPLE
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G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z-sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t-12*z^3-6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4+z^2*t^2+6*z^6*t^2-4*z^6*t^3-4*z^6*t+z^6*t^4+2*t*z+11*z^2)): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 14 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A077587, A000108, A114463, A114464, A114465, A102402.
Sequence in context: A015698 A068587 A001058 this_sequence A092569 A088573 A126388
Adjacent sequences: A114459 A114460 A114461 this_sequence A114463 A114464 A114465
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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), Nov 29 2005
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