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 A159041 Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments. 11
 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 Roger L. Bagula, Another Mathematica program for A159041. EXAMPLE Triangle begins as follows:   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; 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 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 January 21 21:53 EST 2022. Contains 350480 sequences. (Running on oeis4.)