OFFSET
0,4
COMMENTS
a(n) is found in row n, column 0, of triangle A136467 for n>=0.
For n > 0, number of increasing integer sequences 1 <= a_1 < ... < a_n <= 2^(n-1). - Charles R Greathouse IV, Aug 08 2010
The (n-1)-dimensional hypercube has 2^(n-1) corners. There are binomial(2^(n-1),n) ways of selecting a set of n corners. So a(n) is the number of simplices (hyper-tetrahedra) with vertices defined by a corner subset of a (n-1)-dimensional hypercube. (This count includes degenerate polytopes with zero volume.) - R. J. Mathar, Jan 16 2016
LINKS
J. Brandts, S. Dijkhuis, V. de Haan, M. Krizek, There are only two nonobtuse binary triangulations of the unit n-cube, Comp. Geom. 46 (2013) 286-297, Table 1.
FORMULA
a(n) = [x^n] Sum_{i>=0} (1/2)^i * log(1 + 2^i*x)^i/i!.
O.g.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n!*2^n). - Paul D. Hanna, Sep 26 2010
a(n) ~ 2^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016
EXAMPLE
From Paul D. Hanna, Sep 26 2010: (Start)
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 70*x^4 + 4368*x^5 +...
A(x) = 1 + log(1+2*x)/2 + log(1 + 2^2*x)^2/(2!*2^2) + log(1 + 2^3*x)^3/(3!*2^3) + log(1 + 2^4*x)^4/(4!*2^4) +... (End)
MATHEMATICA
Table[Binomial[2^(n-1), n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
PROG
(PARI) {a(n)=binomial(2^(n-1), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = Coefficient of x^k in series: */
{a(n)=polcoeff(sum(i=0, n, (1/2)^i*log(1+2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, log(1+2^m*x+x*O(x^n))^m/(m!*2^m)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 26 2010
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Dec 31 2007
STATUS
approved