A136465 - OEIS

%I #24 Jul 31 2016 13:43:47

%S 1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,

%T 312268282598377321216,30813235422145714150738944,

%U 11005261717918037175659349191168,14391972654784168932973746746691440640,69538271351155829150354851003285125277716480,1250303357941919088313448625534941836891635347865600

%N Row 0 of square array A136462: a(n) = C(2^(n-1), n) for n>=0.

%C a(n) is found in row n, column 0, of triangle A136467 for n>=0.

%C For n > 0, number of increasing integer sequences 1 <= a_1 < ... < a_n <= 2^(n-1). - _Charles R Greathouse IV_, Aug 08 2010

%C 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

%H J. Brandts, S. Dijkhuis, V. de Haan, M. Krizek, <a href="http://dx.doi.org/10.1016/j.comgeo.2012.09.005">There are only two nonobtuse binary triangulations of the unit n-cube</a>, Comp. Geom. 46 (2013) 286-297, Table 1.

%F a(n) = [x^n] Sum_{i>=0} (1/2)^i * log(1 + 2^i*x)^i/i!.

%F O.g.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n!*2^n). - _Paul D. Hanna_, Sep 26 2010

%F a(n) ~ 2^(n*(n-1)) / n!. - _Vaclav Kotesovec_, Jul 02 2016

%e From _Paul D. Hanna_, Sep 26 2010: (Start)

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 70*x^4 + 4368*x^5 +...

%e 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)

%t Table[Binomial[2^(n-1),n], {n,0,15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)

%o (PARI) {a(n)=binomial(2^(n-1),n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) /* a(n) = Coefficient of x^k in series: */

%o {a(n)=polcoeff(sum(i=0,n,(1/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(sum(m=0,n,log(1+2^m*x+x*O(x^n))^m/(m!*2^m)),n)}

%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 26 2010

%Y Cf. A136462; other rows: A014070, A136466, A101346; A136467.

%K nonn,easy

%O 0,4

%A _Paul D. Hanna_, Dec 31 2007