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Search: id:A108443
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| A108443 |
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Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k triple descents (i.e. ddd's). |
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+0
2
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| 1, 2, 6, 3, 1, 21, 24, 15, 5, 1, 80, 150, 145, 84, 31, 7, 1, 322, 857, 1145, 949, 528, 202, 53, 9, 1, 1347, 4692, 8096, 8801, 6598, 3551, 1394, 398, 81, 11, 1, 5798, 25102, 53457, 72338, 68594, 47805, 25092, 10019, 3040, 692, 115, 13, 1, 25512, 132484, 337132
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n has 2n-1 terms (n>=1). Row sums yield A027307. Column 0 yields A106228.
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REFERENCES
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Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
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G.f.=G=G(t, z) satisfies G=1+z(t+z-tz)^2*G^3+z(2-t)(t+z-tz)G^2+2z(1-t)G.
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EXAMPLE
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T(2,1)=3 because we have uUddd, Uuddd and UdUddd.
Triangle begins:
1;
2;
6,3,1;
21,24,15,5,1;
80,150,145,84,31,7,1;
322,857,1145,949,528,202,53,9,1;
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PROGRAM
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(PARI) {T(n, k)=local(G=1+z*O(z^n)+t*O(t^k)); for(k=1, n, G=1+z*(t+z-t*z)^2*G^3+z*(2-t)*(t+z-t*z)*G^2+2*z*(1-t)*G); polcoeff(polcoeff(G, n, z), k, t)}
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CROSSREFS
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Cf. A027307, A106228.
Sequence in context: A136694 A164104 A050138 this_sequence A011042 A136758 A056113
Adjacent sequences: A108440 A108441 A108442 this_sequence A108444 A108445 A108446
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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) and Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2005
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