|
%I #19 Feb 17 2022 01:08:24
%S 1,2,5,15,51,204,933,5115,32385,245214,2090961,20648547,221781915,
%T 2679261840,34419818241,488332067679,7271691132033,118162937240730,
%U 1998172269602685,36552556172242359,691550102624919651,14056929989746659252
%N Extreme points of set of n X n symmetric substochastic matrices.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.25(b).
%H Vincenzo Librandi, <a href="/A053553/b053553.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: ((1+x)/(1-x))^(1/4) * exp(x*(3 + x)/2).
%F a(n) ~ n! * 2^(-1/4)*Gamma(3/4)*exp(2)/(Pi*n^(3/4)). - _Vaclav Kotesovec_, Aug 13 2013
%F Recurrence: 2*a(n) = 4*a(n-1) + 2*(n-1)^2*a(n-2) - 3*(n-2)*(n-1)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Aug 13 2013
%t max = 30; f[x_]:= ((1+x)/(1-x))^(1/4)*Exp[x*(3+x)/2]; CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Nov 30 2011 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( ((1+x)/(1-x))^(1/4)*exp(x*(3 + x)/2) )) \\ _G. C. Greubel_, May 16 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( ((1+x)/(1-x))^(1/4)*Exp(x*(3+x)/2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 16 2019
%o (Sage) m = 30; T = taylor(((1+x)/(1-x))^(1/4)*exp(x*(3+x)/2), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 16 2019
%Y Cf. A006847.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, Jan 16 2000
|
