A159041 - OEIS

%I #53 Mar 18 2022 18:41:32

%S 1,1,1,1,-10,1,1,-25,-25,1,1,-56,246,-56,1,1,-119,1072,1072,-119,1,1,

%T -246,4047,-11572,4047,-246,1,1,-501,14107,-74127,-74127,14107,-501,1,

%U 1,-1012,46828,-408364,901990,-408364,46828,-1012,1,1,-2035,150602,-2052886,7685228,7685228,-2052886,150602,-2035,1

%N Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.

%C 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.

%C 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 ).

%C For example,

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

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

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

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

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

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

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

%C 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

%C 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 ).

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

%C 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).

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

%C The following table lists various sequences constructed using this transform:

%C Parameter Triangle  Triangle  Odd-numbered

%C m             U         T        rows

%C 0          A007312  A007312   A034870

%C 1          A008292  A159041   A171692

%C 2          A060187  A225356   A225076

%C 3          A142458  A225433   A225398

%C 4          A142459  A225434   A225415

%H G. C. Greubel, <a href="/A159041/b159041.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Roger L. Bagula, <a href="/A159041/a159041.txt">Another Mathematica program for A159041</a>.

%F T(n, k) = T(n, k-1) + (-1)^k*A008292(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - _R. J. Mathar_, May 08 2013

%e Triangle begins as follows:

%e   1;

%e   1,     1;

%e   1,   -10,      1;

%e   1,   -25,    -25,        1;

%e   1,   -56,    246,      -56,       1;

%e   1,  -119,   1072,     1072,    -119,       1;

%e   1,  -246,   4047,   -11572,    4047,    -246,        1;

%e   1,  -501,  14107,   -74127,  -74127,   14107,     -501,      1;

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

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

%p 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:

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

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

%p 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 )) );

%p end;

%p U:=A008292;

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

%p A159041 := proc(n, k)

%p     if k = 0 then

%p         1;

%p     elif k <= floor(n/2) then

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

%p     else

%p         A159041(n, n-k) ;

%p     end if;

%p end proc: # _R. J. Mathar_, May 08 2013

%t A[n_, 1] := 1;

%t A[n_, n_] := 1;

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

%t 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);

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

%t Flatten[%]

%o (Sage)

%o def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )

%o @CachedFunction

%o def T(n,k):

%o     if (k==0 or k==n): return 1

%o     elif (k <= (n//2)): return T(n,k-1) + (-1)^k*A008292(n+2,k+1)

%o     else: return T(n,n-k)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022

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

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 03 2009

%E Edited by _N. J. A. Sloane_, May 07 2013, May 11 2013