OFFSET
1,9
COMMENTS
Because in the case n=2 these numbers are the classical signed tangent numbers (A000182) we call T(n, k) also 'generalized tangent numbers' when studied in connection with generalized Bernoulli numbers.
FORMULA
T(n, k) is the derivative of OmegaPolynomial(n, k) evaluated at x = 0.
EXAMPLE
[n\k][0 1 2 3 4 5 ...]
------------------------------------------------------------------
[1] 0, 1, 0, 0, 0, 0, ... [A063524]
[2] 0, 1, -2, 16, -272, 7936, ... [A000182]
[3] 0, 1, -9, 477, -74601, 25740261, ... [A293951]
[4] 0, 1, -34, 11056, -14873104, 56814228736, ... [A273352]
[5] 0, 1, -125, 249250, -2886735625, 122209131374375, ... [A318258]
[6] 0, 1, -461, 5699149, -574688719793, 272692888959243481, ...
[A010763]
MAPLE
# Prints square array row-wise. The function OmegaPolynomial is defined in A318146.
for n from 1 to 6 do seq(coeff(OmegaPolynomial(n, k), x, 1), k=0..6) od;
# In the sequence format:
0, seq(seq(coeff(OmegaPolynomial(n-k+1, k), x, 1), k=0..n), n=1..9);
# Alternatively, based on the recurrence of the André numbers:
ANum := proc(m, n) option remember; if n = 0 then return 1 fi;
`if`(modp(n, m) = 0, -1, 1); [seq(m*k, k=0..(n-1)/m)];
%%*add(binomial(n, k)*ANum(m, k), k in %) end:
TNum := proc(n, k) if k=1 then 1 elif k=0 or n=1 then 0 else ANum(n, n*k-1) fi end:
for n from 1 to 6 do seq(TNum(n, k), k = 0..6) od;
MATHEMATICA
OmegaPolynomial[m_, n_] := Module[{S}, S = Series[MittagLefflerE[m, z]^x, {z, 0, 10}]; Expand[(m*n)! Coefficient[S, z, n]]];
T[n_, k_] := D[OmegaPolynomial[n, k], x] /. x -> 0;
Table[T[n - k, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 27 2023 *)
PROG
(Sage)
# Prints the array row-wise. The function OmegaPolynomial is in A318146.
for m in (1..6):
print([0] + [list(OmegaPolynomial(m, n))[1] for n in (1..6)])
# Alternatively, based on the recurrence of the André numbers:
@cached_function
def ANum(m, n):
if n == 0: return 1
t = [m*k for k in (0..(n-1)//m)]
s = sum(binomial(n, k)*ANum(m, k) for k in t)
return -s if m.divides(n) else s
def TNum(m, n):
if n == 1: return 1
if n == 0 or m == 1: return 0
return ANum(m, m*n-1)
for m in (1..6): print([TNum(m, n) for n in (0..6)])
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 22 2018
STATUS
approved