A116078 Column 0 of triangle A116077.

1, 2, 7, 28, 117, 496, 2110, 8968, 38017, 160648, 676626, 2840872, 11892562, 49650368, 206773372, 859186768, 3562780057, 14746127608, 60929182282, 251358948328, 1035479267542, 4260071237728, 17505144024292

Offset

0,2

Comments

a(n) equals the number of sequences (x(1),...,x(n)) of n numbers from {0,1,...,n} such that x(i+1) <= x(i)+1 for i=1,...,n-1 and x(1) <= x(n)+1. This is proved in a linked pdf, as well as another interpretation as the downward closed subsets of a certain poset. - Clayton Thomas, Jul 16 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Clayton Thomas, A note on the cylinder poset

Formula

G.f.: A(x) = 1/sqrt(1-4*x) + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2).

a(n) = (n+3)*C(2*n-1,n) - 2^(2*n-1), a(n) ~ 2^(2*n - 1) * sqrt(n) / sqrt(Pi). - Vaclav Kotesovec, Oct 28 2012

Conjecture: a(n) = 2^(2*n)*(Sum_{j=1..n+2-floor((n+3)/2)} (cos(j*Pi/(n+3)))^(2*n)). - L. Edson Jeffery, Nov 23 2013

Mathematica

Flatten[{1, Table[(n+3)*Binomial[2*n-1, n]-2^(2*n-1), {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/sqrt(1-4*X)+4*X^2/(1+sqrt(1-4*X))^2/(1-4*X)^(3/2), n, x)}

Crossrefs

Cf. A116077.

Sequence in context: A128611 A061539 A232970 * A150647 A150648 A150649

Adjacent sequences: A116075 A116076 A116077 * A116079 A116080 A116081

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2006

Status

approved