%I #15 Nov 22 2018 12:35:01

%S 1,1,1,2,6,6,36,6,24,240,120,120,1800,1800,120,720,15120,25200,5040,

%T 5040,141120,352800,141120,5040,40320,1451520,5080320,3386880,362880,

%U 362880,16329600,76204800,76204800,16329600,362880,3628800,199584000,1197504000,1676505600,598752000,39916800

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

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

%C 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).

%H M. V. Koutras, <a href="http://www.fq.math.ca/Scanned/32-1/koutras.pdf">Eulerian numbers associated with sequences of polynomials</a>, The Fibonacci Quarterly, 32 (1994), 44-57.

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

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

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

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

%F 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! + ....

%F 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.

%F Row sums 2^n*n! = A000165.

%e Table begins

%e n\k| 0 1 2 3 4

%e = = = = = = = = = = = = = = = = = = = = =

%e 0 | 1

%e 1 | 1 1

%e 2 | 2 6

%e 3 | 6 36 6

%e 4 | 24 240 120

%e 5 | 120 1800 1800 120

%e 6 | 720 15120 25200 5040

%e 7 | 5040 141120 352800 141120 5040

%e 8 | 40320 1451520 5080320 3386880 362880

%e ...

%e Row 3: (6 + 36*t + 6*t^2)/(1 - t)^4 = 1*2*3 + 3*4*5*t + 5*6*7*t^2 + ....

%e 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 + ....

%p #A228955

%p for n from 0 to 10 do

%p seq(n!*binomial(n+1,2*k), k = 0..floor((n+1)/2))

%p end do;

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

%Y Cf. A000165 (row sums), A131980.

%K nonn,easy,tabf

%O 0,4

%A _Peter Bala_, Sep 09 2013