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A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

LINKS

Table of n, a(n) for n=0..35.

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

FORMULA

exp[b(.,t)*x] = 1 - 2x - (t-1) * x^2; therefore exp[a(.,t)*x] = 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)

EXAMPLE

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;

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 *)

PROG

(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

(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

(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

(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

CROSSREFS

Cf. A228955.

Sequence in context: A096485 A125032 A076743 * A217448 A280705 A027760

Adjacent sequences:  A131977 A131978 A131979 * A131981 A131982 A131983

KEYWORD

easy,nonn,tabf

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified February 16 22:46 EST 2020. Contains 331976 sequences. (Running on oeis4.)