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A159041 Triangle read by rows: row n (n>=0) gives coefficients of polynomial p(n,x) of degree n defined in comments. 10
1, 1, 1, 1, -10, 1, 1, -25, -25, 1, 1, -56, 246, -56, 1, 1, -119, 1072, 1072, -119, 1, 1, -246, 4047, -11572, 4047, -246, 1, 1, -501, 14107, -74127, -74127, 14107, -501, 1, 1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1, 1, -2035, 150602 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Let E(n,k) (1 <= k <= n) denote the Eulerian numbers as defined in A008292. Then we define polynomials p(n,x) for n >= 0 as follows.

p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*E(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*E(n+2,k+1)*x^k ).

For example,

p(0,x) = (1-x)/(1-x) = 1,

p(1,x) = (1-x^2)/(1-x) = 1 + x,

p(2,x) = (1 - 11*x^ + 11*x^2 - x^3)/(1-x) = 1 - 10*x + x^2,

p(3,x) = (1 - 26*x + 26*x^3 - x^4)/(1-x) = 1 - 25*x - 25*x^2 + x^3),

p(4,x) = (1 - 57*x + 302*x^2 - 302*x^3 + 57*x^3 + x^5)/(1-x)

       = 1 - 56*x + 246*x^2 - 56*x^3 + x^4.

Row sums are: {1, 2, -8, -48, 136, 1908, -3968, -121040, 176896, 11561820, -11184128, ...}.

More generally, there is a triangle-to-triangle transformation U -> T defined as follows.

Let U(n,k) (1 <= k <= n) be a triangle of nonnegative numbers in which the rows are symmetric about the middle. Define polynomials p(n,x) for n >= 0 by

p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*U(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*U(n+2,k+1)*x^k ).

The n-th row of the new triangle T(n,k) (0 <= k <= n) gives the coefficients in the expansion of p(n+2).

The new triangle may be defined recursively by: T(n,0)=1; T(n,k) = T(n,k-1) + (-1)^k*U(n+2,k) for 1 <= k <= floor(n/2); T(n,k) = T(n,n-k).

Note that the central terms in the odd-numbered rows of U(n,k) do not get used.

The following table lists various sequences constructed using this transform:

Parameter Triangle  Triangle  Odd-numbered

m             U         T        rows

0          A007312  A007312   A034870

1          A008292  A159041   A171692

2          A060187  A225356   A225076

3          A142458  A225433   A225398

4          A142459  A225434   A225415

LINKS

Table of n, a(n) for n=0..47.

Roger L. Bagula, Another Mathematica program for A159041

EXAMPLE

1;

1, 1;

1, -10, 1;

1, -25, -25, 1;

1, -56, 246, -56, 1;

1, -119, 1072, 1072, -119, 1;

1, -246, 4047, -11572, 4047, -246, 1;

1, -501, 14107, -74127, -74127, 14107, -501, 1;

1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1;

1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1;

1, -4082, 474189, -9713496, 56604978, -105907308, 56604978, -9713496, 474189, -4082, 1;

MAPLE

A008292 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc:

# row n of new triangle T(n, k) in terms of old triangle U(n, k):

p:=proc(n) local k; global U;

simplify( (1/(1-x)) * ( add((-1)^k*U(n+2, k+1)*x^k, k=0..floor(n/2)) + add((-1)^(n+k)*U(n+2, k+1)*x^k, k=ceil((n+2)/2)..n+1 )) );

end;

U:=A008292;

for n from 0 to 6 do lprint(simplify(p(n))); od: # N. J. A. Sloane, May 11 2013

A159041 := proc(n, k)

    if k = 0 then

        1;

    elif k <= floor(n/2) then

        A159041(n, k-1)+(-1)^k*A008292(n+2, k+1) ;

    else

        A159041(n, n-k) ;

    end if;

end proc: # R. J. Mathar, May 08 2013

MATHEMATICA

Clear[A, p, n, i];

A[n_, 1] := 1;

A[n_, n_] := 1;

A[n_, k_] := (n - k + 1)A[n - 1, k - 1] + k A[n - 1, k];

p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);

Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];

Flatten[%]

CROSSREFS

Cf. A007312, A008292, A034870, A060187, A142458, A142459, A159041, A171692, A225076, A225356, A225398, A225415, A225433, A225434.

Sequence in context: A202941 A166341 A113280 * A154979 A146765 A190152

Adjacent sequences:  A159038 A159039 A159040 * A159042 A159043 A159044

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Apr 03 2009

EXTENSIONS

Edited by N. J. A. Sloane, May 07 2013, May 11 2013

STATUS

approved

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Last modified February 16 21:37 EST 2020. Contains 331975 sequences. (Running on oeis4.)