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A081658
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Triangle read by rows: T(n, k) = (-2)^k*binomial(n, k)*Euler(k, 1/2).
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7
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1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 5, 1, 0, -10, 0, 25, 0, 1, 0, -15, 0, 75, 0, -61, 1, 0, -21, 0, 175, 0, -427, 0, 1, 0, -28, 0, 350, 0, -1708, 0, 1385, 1, 0, -36, 0, 630, 0, -5124, 0, 12465, 0, 1, 0, -45, 0, 1050, 0, -12810, 0, 62325, 0, -50521, 1, 0, -55, 0, 1650, 0, -28182, 0, 228525, 0, -555731, 0, 1, 0, -66, 0, 2475, 0
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OFFSET
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0,9
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COMMENTS
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Nonzero diagonals of the triangle are of the form A000364(k)*binomial(n+2k,2k)*(-1)^k.
A363393 is the dual triangle ('dual' in the sense of Euler-tangent versus Euler-secant numbers). - Peter Luschny, Jun 05 2023
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LINKS
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FORMULA
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Coefficients of the polynomials in k in the binomial transform of the expansion of 2/(exp(kx)+exp(-kx)).
p{n}(0) = Signed Euler secant numbers A122045.
p{n}(1) = Signed Euler tangent numbers A155585.
p{n}(2) has e.g.f. 2*exp(x)/(exp(-2*x)+1) A119880.
2^n*p{n}(1/2) = Signed Springer numbers A188458.
3^n*p{n}(1/3) has e.g.f. 2*exp(4*x)/(exp(6*x)+1)
4^n*p{n}(1/4) has e.g.f. 2*exp(5*x)/(exp(8*x)+1).
The GCD of the rows without the first column: A155457. (End)
T(n, k) = [x^(n - k)] Euler(k) / (1 - x)^(k + 1).
For a recursion see the Python program.
Conjecture: If n is prime then n divides T(n, k) for 1 <= k <= n-1. (End)
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EXAMPLE
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The triangle begins
[0] 1;
[1] 1, 0;
[2] 1, 0, -1;
[3] 1, 0, -3, 0;
[4] 1, 0, -6, 0, 5;
[5] 1, 0, -10, 0, 25, 0;
[6] 1, 0, -15, 0, 75, 0, -61;
[7] 1, 0, -21, 0, 175, 0, -427, 0;
...
The triangle shows the coefficients of the following polynomials:
[1] 1;
[2] 1 - x^2;
[3] 1 - 3*x^2;
[4] 1 - 6*x^2 + 5*x^4;
[5] 1 - 10*x^2 + 25*x^4;
[6] 1 - 15*x^2 + 75*x^4 - 61*x^6;
[7] 1 - 21*x^2 + 175*x^4 - 427*x^6;
...
These polynomials are the permanents of the n X n matrices with all entries above the main antidiagonal set to 'x' and all entries below the main antidiagonal set to '-x'. The main antidiagonals consist only of ones. Substituting x <- 1 generates the Euler tangent numbers A155585. (Compare with A046739.)
(End)
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MAPLE
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ogf := n -> euler(n) / (1 - x)^(n + 1):
ser := n -> series(ogf(n), x, 16):
T := (n, k) -> coeff(ser(k), x, n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Jun 05 2023
T := (n, k) -> (-2)^k*binomial(n, k)*euler(k, 1/2):
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Apr 03 2024
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MATHEMATICA
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sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n - k), {k, 0, n}];
Table[CoefficientList[sk[n, x], x] // Reverse, {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 04 2019 *)
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PROG
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(Sage)
R = PolynomialRing(ZZ, 'x')
@CachedFunction
def p(n, x) :
if n == 0 : return 1
return add(p(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A081658_row(n) : return [R(p(n, x)).reverse()[i] for i in (0..n)]
(Python)
from functools import cache
@cache
def T(n: int, k: int) -> int:
if k == 0: return 1
if k % 2 == 1: return 0
if k == n: return -sum(T(n, j) for j in range(0, n - 1, 2))
return (T(n - 1, k) * n) // (n - k)
for n in range(10):
print([T(n, k) for k in range(n + 1)]) # Peter Luschny, Jun 05 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Error in data corrected and new name by Peter Luschny, Apr 03 2024
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STATUS
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approved
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