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%I #16 Nov 27 2023 06:14:35
%S 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,
%T -74601,7936,0,0,1,-461,249250,-14873104,25740261,-353792,0,0,1,-1715,
%U 5699149,-2886735625,56814228736,-16591655817,22368256,0,0,1,-6434,132908041,-574688719793,122209131374375,-495812444583424,17929265150637,-1903757312,0
%N 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.
%C 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.
%F T(n, k) is the derivative of OmegaPolynomial(n, k) evaluated at x = 0.
%F 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).
%e [n\k][0 1 2 3 4 5 ...]
%e ------------------------------------------------------------------
%e [1] 0, 1, 0, 0, 0, 0, ... [A063524]
%e [2] 0, 1, -2, 16, -272, 7936, ... [A000182]
%e [3] 0, 1, -9, 477, -74601, 25740261, ... [A293951]
%e [4] 0, 1, -34, 11056, -14873104, 56814228736, ... [A273352]
%e [5] 0, 1, -125, 249250, -2886735625, 122209131374375, ... [A318258]
%e [6] 0, 1, -461, 5699149, -574688719793, 272692888959243481, ...
%e [A010763]
%p # Prints square array row-wise. The function OmegaPolynomial is defined in A318146.
%p for n from 1 to 6 do seq(coeff(OmegaPolynomial(n, k), x, 1), k=0..6) od;
%p # In the sequence format:
%p 0, seq(seq(coeff(OmegaPolynomial(n-k+1, k), x, 1), k=0..n), n=1..9);
%p # Alternatively, based on the recurrence of the André numbers:
%p ANum := proc(m, n) option remember; if n = 0 then return 1 fi;
%p `if`(modp(n, m) = 0, -1, 1); [seq(m*k, k=0..(n-1)/m)];
%p %%*add(binomial(n, k)*ANum(m, k), k in %) end:
%p 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:
%p for n from 1 to 6 do seq(TNum(n, k), k = 0..6) od;
%t OmegaPolynomial[m_, n_] := Module[{S}, S = Series[MittagLefflerE[m, z]^x, {z, 0, 10}]; Expand[(m*n)! Coefficient[S, z, n]]];
%t T[n_, k_] := D[OmegaPolynomial[n, k], x] /. x -> 0;
%t Table[T[n - k, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Nov 27 2023 *)
%o (Sage)
%o # Prints the array row-wise. The function OmegaPolynomial is in A318146.
%o for m in (1..6):
%o print([0] + [list(OmegaPolynomial(m, n))[1] for n in (1..6)])
%o # Alternatively, based on the recurrence of the André numbers:
%o @cached_function
%o def ANum(m, n):
%o if n == 0: return 1
%o t = [m*k for k in (0..(n-1)//m)]
%o s = sum(binomial(n, k)*ANum(m, k) for k in t)
%o return -s if m.divides(n) else s
%o def TNum(m, n):
%o if n == 1: return 1
%o if n == 0 or m == 1: return 0
%o return ANum(m, m*n-1)
%o for m in (1..6): print([TNum(m, n) for n in (0..6)])
%Y Cf. A318146, A181937, A063524, A000182, A293951, A273352, A318258.
%K sign,tabl
%O 1,9
%A _Peter Luschny_, Aug 22 2018
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