A318253 Coefficient of x of the OmegaPolynomials (A318146), T(n, k) = [x] P(n, k) with n>=1 and k>=0, square array read by ascending antidiagonals.

0, 0, 1, 0, 1, 0, 0, 1, -2, 0, 0, 1, -9, 16, 0, 0, 1, -34, 477, -272, 0, 0, 1, -125, 11056, -74601, 7936, 0, 0, 1, -461, 249250, -14873104, 25740261, -353792, 0, 0, 1, -1715, 5699149, -2886735625, 56814228736, -16591655817, 22368256, 0, 0, 1, -6434, 132908041, -574688719793, 122209131374375, -495812444583424, 17929265150637, -1903757312, 0

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.

Links

Table of n, a(n) for n=1..55.

Formula

T(n, k) is the derivative of OmegaPolynomial(n, k) evaluated at x = 0.

Apart from the border cases n=1 and k=0 the generalized tangent numbers are a subset of the André numbers A181937; more precisely: T(n, k) is 1 if k = 1 else if k = 0 or n = 1 then T(n, k) = 0 else T(n,k) = (-1)^(n+1)*A181937(n, n*k-1).

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

Cf. A318146, A181937, A063524, A000182, A293951, A273352, A318258.

Sequence in context: A367000 A289084 A193033 * A249772 A193531 A093492

Adjacent sequences: A318250 A318251 A318252 * A318254 A318255 A318256

Keyword

sign,tabl

Author

Peter Luschny, Aug 22 2018

Status

approved