A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330.

1, 2, 6, 2, 24, 24, 120, 240, 24, 720, 2400, 720, 5040, 25200, 15120, 720, 40320, 282240, 282240, 40320, 362880, 3386880, 5080320, 1451520, 40320, 3628800, 43545600, 91445760, 43545600, 3628800, 39916800, 598752000, 1676505600, 1197504000, 199584000, 3628800

Offset: 0

Author

Tom Copeland, Oct 30 2007, Nov 29 2007, Nov 30 2007

Keywords

• easy
• nonn
• tabf

Comments

Construct the infinite array of polynomials

a(0,t) = 1

a(1,t) = 2

a(2,t) = 6 + 2 t

a(3,t) = 24 + 24 t

a(4,t) = 120 + 240 t + 24 t^2

a(5,t) = 720 + 2400 t + 720 t^2

a(6,t) = 5040 + 25200 t + 15120 t^2 + 720 t^3

This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.

b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2.

Then A000165(n) = a(n,1).

Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2).

n! * A000129(n+1) = a(n,2) = A110327(n,0).

A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,2).

A000142(n+1) = a(n,0).

From Peter Bala, Sep 09 2013: (Start)

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 the present table is the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x). See A228955 for the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x + 1). (End)

Examples

			Triangle begins as:
        1;
        2;
        6,        2;
       24,       24;
      120,      240,       24;
      720,     2400,      720;
     5040,    25200,    15120,      720;
    40320,   282240,   282240,    40320;
   362880,  3386880,  5080320,  1451520,   40320;
  3628800, 43545600, 91445760, 43545600, 3628800;
		

Links

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

Crossrefs

Cf. A228955.

Programs

GAP

Flat(List([0..10], n-> List([0..Int(n/2)], k-> Factorial(n)*Binomial(n+1, 2*k+1) ))); # G. C. Greubel, Dec 30 2019

Magma

[Factorial(n)*Binomial(n+1, 2*k+1): k in [0..Floor(n/2)], n in [0..10]]; // G. C. Greubel, Dec 30 2019

Maple

for n from 0 to 10 do
seq( n!*binomial(n+1,2*k+1), k = 0..floor(n/2) )
end do; # Peter Bala, Sep 09 2013

Mathematica

Table[n!*Binomial[n+1, 2*k+1], {n,0,10}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Dec 30 2019 *)

PARI

T(n,k) = n!*binomial(n+1, 2*k+1);
for(n=0,10, for(k=0, n\2, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 30 2019

Sage

[[factorial(n)*binomial(n+1, 2*k+1) for k in (0..floor(n/2))] for n in (0..10)] # G. C. Greubel, Dec 30 2019

Formula

E.g.f. for the polynomials b(.,t), introduced above, is 1 - 2x - (t-1) * x^2; therefore e.g.f. for the polynomials a(.,t), which are the row polynomials of this array, is 1 / ( 1 - 2x - (t-1) * x^2 ) = (t-1) / ( t - ( 1 + x*(t-1) )^2 ).

Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. at u = 1/t. Compare A076743.

a(n,t) = n!*Sum_{k>=0} binomial(n+1,2k+1) * t^k = n!*Sum_{k>=0} A034867(n,k) * t^k.

Additional relations are given by formulas in A133314.

From Peter Bala, Sep 09 2013: (Start)

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

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

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

The row polynomial a(n,t) satisfies t*a(n,t)/(1 - t)^(n+1) = Sum_{j>=1} P(n,2*j)*t^j. For example, for n = 3 we have t*(24 + 24*t)/(1 - t)^4 = 2*3*4*t + (4*5*6)*t^2 + (6*7*8)*t^3 + ..., while for n = 4 we have t*(120 + 240*t + 24*t^2)/(1 - t)^5 = (2*3*4*5)*t + (4*5*6*7)*t^2 + (6*7*8*9)*t^3 + .... (End)

Extensions

Removed erroneous and duplicate statements. - Tom Copeland, Dec 03 2013