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Search: id:A129528
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| A129528 |
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Number of Dyck paths such that the sum of the peak-abscissae is n. |
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2
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| 1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 19, 24, 30, 39, 48, 60, 75, 93, 115, 142, 173, 213, 260, 316, 383, 465, 560, 676, 812, 974, 1165, 1393, 1658, 1975, 2345, 2779, 3288, 3887, 4582, 5398, 6346, 7452, 8735, 10230, 11956, 13964, 16283, 18964, 22057
(list; graph; listen)
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OFFSET
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0,5
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
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FORMULA
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Column sums of triangle A129174. The generating polynomial for row n of A129174 is P[n](t) = t^n*binom[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binom[2n,n] is a Gaussian polynomial (in t).
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EXAMPLE
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a(6)=3 because we have (i) UUDUDD with peak-abscissae 2,4; (ii) UDUUUDDD with peak-abscissae 1,5; and (iii) UUUUUUDDDDDD with peak-abscissa 6 (here U=(1,1) and D=(1,-1)).
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MAPLE
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br:=n->sum(q^i, i=0..n-1): f:=n->product(br(j), j=1..n): cbr:=(n, k)->f(n)/f(k)/f(n-k): P:=n->sort(expand(simplify(q^n*cbr(2*n, n)/br(n+1)))): seq(add(coeff(P(m), q, l), m=0..l), l=0..60);
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CROSSREFS
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Cf. A129174.
Sequence in context: A003073 A123946 A002569 this_sequence A052336 A061287 A064651
Adjacent sequences: A129525 A129526 A129527 this_sequence A129529 A129530 A129531
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2007
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