%I #16 Jul 27 2018 04:39:10

%S 1,2,2,8,20,12,48,224,344,168,384,2880,8096,9872,4272,3840,42240,

%T 186816,407936,430688,171168,46080,698880,4451328,15030528,27944576,

%U 26627648,9915072,645120,12902400,111605760,535271424,1519126272

%N Coefficients of polynomials D(n,x) related to median Euler numbers.

%C 2^n(x+1) divides D(n,x).

%H A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26.

%F Recurrence: D(0, x)=1, D(n, x) = (x+1)(x+2)D(n-1, x+2) - x(x+1)D(n-1, x).

%F G.f.: Sum[n>=0, D(n, x)t^n] = 1/(1-2(x+1)t/(1-2(x+2)t/(1-4(x+3)t/(1-4(x+4)t/...)))).

%F G.f.: Sum_{n>=0} D(n,y)*x^n = Sum_{n>=0} n!*(2*x)^n*Product_{k=1..n} (k+y)/(1+k*(k+1)*x). - _Paul D. Hanna_, Sep 05 2012

%e D(0,x) = 1,

%e D(1,x) = 2*x + 2,

%e D(2,x) = 8*x^2 + 20*x + 12,

%e D(3,x) = 48*x^3 + 224*x^2 + 344*x + 168,

%e D(4,x) = 384*x^4 + 2880*x^3 + 8096*x^2 + 9872*x + 4272.

%t d[0, _] = 1; d[n_, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2] - x(x+1)d[n-1, x];

%t Table[CoefficientList[d[n, x], x] // Reverse, {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jul 27 2018 *)

%o (PARI) D(n,x)=if(n<1,1,(x+1)*(x+2)*D(n-1,x+2)-x*(x+1)*D(n-1,x))

%o (PARI) T(n,k)=local(A=sum(m=0,n,m!*(2*x)^m*prod(j=1,m,(j+y)/(1+j*(j+1)*x +x*O(x^n)))));polcoeff(polcoeff(A,n,x),n-k,y)

%o {for(n=0,8,for(k=0,n,print1(T(n,k),", "));print())} \\ _Paul D. Hanna_, Sep 05 2012

%Y D(n, 1/2) = A002832(n+1), D(n, -1/2) = A000657(n).

%Y D(n, 0)/2^n = A098278(n), D(n, 1)/2^n = A098279(n).

%Y Leading coefficients are A000165. Constant terms are in A098431.

%K nonn,tabl

%O 0,2

%A _Ralf Stephan_, Sep 07 2004