A098277 Coefficients of polynomials D(n,x) related to median Euler numbers.

1, 2, 2, 8, 20, 12, 48, 224, 344, 168, 384, 2880, 8096, 9872, 4272, 3840, 42240, 186816, 407936, 430688, 171168, 46080, 698880, 4451328, 15030528, 27944576, 26627648, 9915072, 645120, 12902400, 111605760, 535271424, 1519126272

Offset: 0

Author

Ralf Stephan, Sep 07 2004

Keywords

• nonn
• tabl

Comments

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

Examples

			D(0,x) = 1,
D(1,x) = 2*x + 2,
D(2,x) = 8*x^2 + 20*x + 12,
D(3,x) = 48*x^3 + 224*x^2 + 344*x + 168,
D(4,x) = 384*x^4 + 2880*x^3 + 8096*x^2 + 9872*x + 4272.
		

Links

• A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.

Crossrefs

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

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

Leading coefficients are A000165. Constant terms are in A098431.

Programs

Mathematica

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];
Table[CoefficientList[d[n, x], x] // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)

PARI

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

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)
{for(n=0,8,for(k=0,n,print1(T(n,k),", "));print())} \\ Paul D. Hanna, Sep 05 2012

Formula

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

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/...)))).

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