

A129528


Number of Dyck paths such that the sum of the peakabscissae 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
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OFFSET

0,5


REFERENCES

G. E. Andrews, The Theory of Partitions, AddisonWesley, 1976.


LINKS



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 peakabscissae 2,4; (ii) UDUUUDDD with peakabscissae 1,5; and (iii) UUUUUUDDDDDD with peakabscissa 6 (here U=(1,1) and D=(1,1)).


MAPLE

br:=n>sum(q^i, i=0..n1): f:=n>product(br(j), j=1..n): cbr:=(n, k)>f(n)/f(k)/f(nk): 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, (1x^(m+k))/(1x^k)+x*O(x^n))), n)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



