OFFSET
0,4
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).
LINKS
M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.
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.
EXAMPLE
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 + ....
MAPLE
MATHEMATICA
Flatten[Table[n!*Binomial[n+1, 2k], {n, 0, 10}, {k, 0, Floor[(n+1)/2]}]](* Harvey P. Dale, Nov 22 2018 *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Peter Bala, Sep 09 2013
STATUS
approved