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A098277 Coefficients of polynomials D(n,x) related to median Euler numbers. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)