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A341724 Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.

%I A341724 #26 Sep 08 2023 13:06:13
%S A341724 1,-2,1,8,-4,1,-50,24,-6,1,416,-200,48,-8,1,-4322,2080,-500,80,-10,1,
%T A341724 53888,-25932,6240,-1000,120,-12,1,-783890,377216,-90762,14560,-1750,
%U A341724 168,-14,1,13031936,-6271120,1508864,-242032,29120,-2800,224,-16,1
%N A341724 Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.
%C A341724 Conjectures from _Mélika Tebni_, Sep 04 2023: (Start)
%C A341724 For 0 < k < p  and p prime, T(p,k) == 0 (mod p).
%C A341724 For 0 <= k < n  and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)
%D A341724 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 A341724 From _Mélika Tebni_, Sep 04 2023: (Start)
%F A341724 E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!).
%F A341724 T(n,k) = (-1)^(n-k)*binomial(n,k)*A000557(n-k).
%F A341724 Recurrence: T(n,0) = (-1)^n*A000557(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
%F A341724 From _Alois P. Heinz_, Sep 04 2023: (Start)
%F A341724 |Sum_{k=0..n} T(n,k)| = A000556(n).
%F A341724 Sum_{k=0..n} |T(n,k)| = A005923(n).
%F A341724 Sum_{k=0..n} k * T(n,k) = A341726(n). (End)
%e A341724 Triangle begins:
%e A341724          1;
%e A341724         -2,        1;
%e A341724          8,       -4,       1;
%e A341724        -50,       24,      -6,       1;
%e A341724        416,     -200,      48,      -8,     1;
%e A341724      -4322,     2080,    -500,      80,   -10,     1;
%e A341724      53888,   -25932,    6240,   -1000,   120,   -12,   1;
%e A341724    -783890,   377216,  -90762,   14560, -1750,   168, -14,   1;
%e A341724   13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1;
%e A341724   ...
%p A341724 egf:= k-> x^k / ((1-2*sinh(-x))*k!):
%p A341724 A341724:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A341724 seq(print(seq(A341724(n,k), k=0..n)), n=0..8); # _Mélika Tebni_, Sep 04 2023
%Y A341724 Column 0 is a signed version of A000557, column 1 is A341727.
%Y A341724 Cf. A000556, A005923, A341723, A341725, A341726.
%K A341724 sign,tabl
%O A341724 0,2
%A A341724 _N. J. A. Sloane_, Mar 04 2021