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

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

Offset

0,5

References

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Paul D. Hanna)

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

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(7/2) * n^(3/2)). - Vaclav Kotesovec, Jun 11 2025

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);

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^k * Product[(1 - x^(k + j))/(1 - x^j), {j, 2, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2025 *)

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