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A129528 Number of Dyck paths such that the sum of the peak-abscissae is n. 3

%I #11 Nov 15 2019 23:18:09

%S 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,

%T 260,316,383,465,560,676,812,974,1165,1393,1658,1975,2345,2779,3288,

%U 3887,4582,5398,6346,7452,8735,10230,11956,13964,16283,18964,22057

%N Number of Dyck paths such that the sum of the peak-abscissae is n.

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

%H Paul D. Hanna, <a href="/A129528/b129528.txt">Table of n, a(n) for n = 0..1000</a>

%F 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).

%F G.f.: Sum_{n>=0} x^n * Product_{k=2..n} (1 - x^(n+k)) / (1 - x^k). - _Paul D. Hanna_, Jan 03 2013

%e 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)).

%p 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);

%o (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)

%o for(n=0,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 03 2013

%Y Cf. A129174.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Apr 20 2007

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Last modified February 22 05:42 EST 2024. Contains 370240 sequences. (Running on oeis4.)