A228955 Table: T(n,k) = n!*binomial(n+1,2*k).

1, 1, 1, 2, 6, 6, 36, 6, 24, 240, 120, 120, 1800, 1800, 120, 720, 15120, 25200, 5040, 5040, 141120, 352800, 141120, 5040, 40320, 1451520, 5080320, 3386880, 362880, 362880, 16329600, 76204800, 76204800, 16329600, 362880, 3628800, 199584000, 1197504000, 1676505600, 598752000, 39916800

Offset: 0

Author

Peter Bala, Sep 09 2013

Keywords

• nonn
• easy
• tabf

Comments

A companion table to A131980(n,k) = n!*binomial(n+1,2*k+1).

Let {P(n,x)}n>=0 be a polynomial sequence. Koutras has defined generalized Eulerian numbers associated with the sequence P(n,x) as the coefficients A(n,k) in the expansion of P(n,x) in a series of factorials of degree n, namely P(n,x) = sum {k = 0..n} A(n,k)* binomial(x+n-k,n). The choice P(n,x) = x^n produces the classical Eulerian numbers of A008292. Let now P(n,x) = x*(x+1)*...*(x+n-1) denote the n-th rising factorial polynomial. Then A131980 is the table of generalized Eulerian numbers associated with the polynomial sequence P(n,2*x) while the present table is the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x+1).

Examples

			Table begins
n\k|    0       1       2       3      4
= = = = = = = = = = = = = = = = = = = = =
0 |     1
1 |     1       1
2 |     2       6
3 |     6      36       6
4 |    24     240     120
5 |   120    1800    1800     120
6 |   720   15120   25200    5040
7 |  5040  141120  352800  141120   5040
8 | 40320 1451520 5080320 3386880 362880
...
Row 3: (6 + 36*t + 6*t^2)/(1 - t)^4 = 1*2*3 + 3*4*5*t + 5*6*7*t^2 + ....
Row 4: (24 + 240*t + 120*t^2)/(1 - t)^5 = 1*2*3*4 + 3*4*5*6*t + 5*6*7*8*t^2 + ....
		

Links

• M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.

Crossrefs

Cf. A000165 (row sums), A131980.

Programs

Maple

#A228955
for n from 0 to 10 do
  seq(n!*binomial(n+1,2*k), k = 0..floor((n+1)/2))
end do;

Mathematica

Flatten[Table[n!*Binomial[n+1,2k],{n,0,10},{k,0,Floor[(n+1)/2]}]](* Harvey P. Dale, Nov 22 2018 *)

Formula

T(n,k) = n!*binomial(n+1,2*k) for n,k >= 0.

Let P(n,x) = x*(x+1)*...*(x+n-1) denote the n-th rising factorial. Then

T(n,k) = sum(j=0..k, (-1)^(k-j)*binomial(n+1,k-j)*P(n,2*j+1) ) for n >= 1.

Recurrence equation: T(n+1,k) = (n+2*k+1)*T(n,k) + (n-2*k+3)*T(n,k-1).

E.g.f.: ( 1 - u*(1 - x) )/( (u - 1)^2 - u^2*x ) = 1 + (1 + x)*u + (2 + 6*x)*u^2/2! + (6 + 36*x + 6*x^2)*u^3/3! + ....

The n-th row polynomial R(n,t) satisfies R(n,t)/(1 - t)^(n+1) = sum(j>=0, P(n,2*j+1)*t^j ). Some examples are given below.

Row sums 2^n*n! = A000165.