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%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
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