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

0,2

Comments

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

Links

Table of n, a(n) for n=0..32.

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

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A168506 A208966 A067640 * A242658 A080040 A060823

Adjacent sequences: A098274 A098275 A098276 * A098278 A098279 A098280

Keyword

nonn,tabl

Author

Ralf Stephan, Sep 07 2004

Status

approved