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A159041
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Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.
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11
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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
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OFFSET
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0,5
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COMMENTS
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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.
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
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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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;
for n from 0 to 6 do lprint(simplify(p(n))); od: # N. J. A. Sloane, May 11 2013
if k = 0 then
1;
elif k <= floor(n/2) then
else
end if;
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MATHEMATICA
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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[%]
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PROG
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(Sage)
def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
@CachedFunction
def T(n, k):
if (k==0 or k==n): return 1
elif (k <= (n//2)): return T(n, k-1) + (-1)^k*A008292(n+2, k+1)
else: return T(n, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022
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CROSSREFS
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Cf. A007312, A008292, A034870, A060187, A142458, A142459, A159041, A171692, A225076, A225356, A225398, A225415, A225433, A225434.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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