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Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.
6

%I #26 Sep 08 2023 13:06:13

%S 1,-2,1,8,-4,1,-50,24,-6,1,416,-200,48,-8,1,-4322,2080,-500,80,-10,1,

%T 53888,-25932,6240,-1000,120,-12,1,-783890,377216,-90762,14560,-1750,

%U 168,-14,1,13031936,-6271120,1508864,-242032,29120,-2800,224,-16,1

%N Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.

%C Conjectures from _Mélika Tebni_, Sep 04 2023: (Start)

%C For 0 < k < p and p prime, T(p,k) == 0 (mod p).

%C For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)

%D Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 3.

%F From _Mélika Tebni_, Sep 04 2023: (Start)

%F E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!).

%F T(n,k) = (-1)^(n-k)*binomial(n,k)*A000557(n-k).

%F Recurrence: T(n,0) = (-1)^n*A000557(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)

%F From _Alois P. Heinz_, Sep 04 2023: (Start)

%F |Sum_{k=0..n} T(n,k)| = A000556(n).

%F Sum_{k=0..n} |T(n,k)| = A005923(n).

%F Sum_{k=0..n} k * T(n,k) = A341726(n). (End)

%e Triangle begins:

%e 1;

%e -2, 1;

%e 8, -4, 1;

%e -50, 24, -6, 1;

%e 416, -200, 48, -8, 1;

%e -4322, 2080, -500, 80, -10, 1;

%e 53888, -25932, 6240, -1000, 120, -12, 1;

%e -783890, 377216, -90762, 14560, -1750, 168, -14, 1;

%e 13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1;

%e ...

%p egf:= k-> x^k / ((1-2*sinh(-x))*k!):

%p A341724:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):

%p seq(print(seq(A341724(n,k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 04 2023

%Y Column 0 is a signed version of A000557, column 1 is A341727.

%Y Cf. A000556, A005923, A341723, A341725, A341726.

%K sign,tabl

%O 0,2

%A _N. J. A. Sloane_, Mar 04 2021