A098278 -id:A098278 - cp's OEIS frontend

A174827 Hankel transform of A098278.

1, 2, 192, 4976640, 115579079884800, 6039552457237856256000000, 1499708022491968274577374576640000000000, 3321055547746756031053448740122923472047308800000000000000

Offset: 0

Paul Barry, Mar 30 2010

Cf. A000178, A174826.

A174827:=n->mul( ((k+1)*(2*k+1)*(2*k+2)*floor((2*k+3)/2))^(n-k), k=0..n): seq(A174827(n), n=0..7); # Wesley Ivan Hurt, Sep 13 2014

Table[Product[((k + 1) (2 k + 1) (2 k + 2) Floor[(2 k + 3)/2])^(n - k), {k, 0, n}], {n, 0, 7}] (* Wesley Ivan Hurt, Sep 13 2014 *)

a(n) = Product{k=0..n, ((k+1)*(2*k+1)*(2*k+2)*floor((2*k+3)/2))^(n-k)}.
A000178(n) divides a(n). - Peter Luschny, Sep 14 2014
a(n) ~ 2^(n*(n+3) + 41/24) * n^(2*n^2 + 7*n/2 + 31/24) * Pi^(3*(n+1)/2) / (A^(5/2) * exp(3*n^2 + 7*n/2 - 5/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019

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

Original entry on oeis.org

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

Ralf Stephan, Sep 07 2004

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

			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.

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

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.

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 *)

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

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

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

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A174827 Hankel transform of A098278.

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

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