OFFSET
0,9
COMMENTS
These are the coefficients of the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 21 2012
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
FORMULA
Coefficients of the polynomials in k in the binomial transform of the expansion of 2/(exp(kx)+exp(-kx)).
From Peter Luschny, Jul 20 2012: (Start)
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)
From Peter Luschny, Jun 05 2023: (Start)
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)
EXAMPLE
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;
...
From Peter Luschny, Sep 17 2021: (Start)
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)
MAPLE
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
MATHEMATICA
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 *)
PROG
(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)]
for n in (0..8) : print(A081658_row(n)) # Peter Luschny, Jul 20 2012
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 26 2003
EXTENSIONS
Typo in data corrected by Peter Luschny, Jul 20 2012
Error in data corrected and new name by Peter Luschny, Apr 03 2024
STATUS
approved