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 A129528 Number of Dyck paths such that the sum of the peak-abscissae is n. 3
 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; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..1000 FORMULA Column sums of triangle A129174. The generating polynomial for row n of A129174 is P[n](t) = t^n*binomial[2n,n]/[n+1], where [n+1] = 1 + t + t^2 + ... + t^n and binomial[2n,n] is a Gaussian polynomial (in t). G.f.: Sum_{n>=0} x^n * Product_{k=2..n} (1 - x^(n+k)) / (1 - x^k). - Paul D. Hanna, Jan 03 2013 EXAMPLE 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)). MAPLE 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); PROG (PARI) a(n)=polcoeff(sum(m=0, n, x^m*prod(k=2, m, (1-x^(m+k))/(1-x^k)+x*O(x^n))), n) for(n=0, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 03 2013 CROSSREFS Cf. A129174. Sequence in context: A003073 A123946 A002569 * A364159 A280200 A052336 Adjacent sequences: A129525 A129526 A129527 * A129529 A129530 A129531 KEYWORD nonn AUTHOR Emeric Deutsch, Apr 20 2007 STATUS approved

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Last modified December 3 09:12 EST 2023. Contains 367539 sequences. (Running on oeis4.)