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

1, -2, 1, 8, -4, 1, -50, 24, -6, 1, 416, -200, 48, -8, 1, -4322, 2080, -500, 80, -10, 1, 53888, -25932, 6240, -1000, 120, -12, 1, -783890, 377216, -90762, 14560, -1750, 168, -14, 1, 13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1

Offset: 0

Author

N. J. A. Sloane, Mar 04 2021

Keywords

• sign
• tabl

Comments

Conjectures from Mélika Tebni, Sep 04 2023: (Start)

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

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

Examples

			Triangle begins:
         1;
        -2,        1;
         8,       -4,       1;
       -50,       24,      -6,       1;
       416,     -200,      48,      -8,     1;
     -4322,     2080,    -500,      80,   -10,     1;
     53888,   -25932,    6240,   -1000,   120,   -12,   1;
   -783890,   377216,  -90762,   14560, -1750,   168, -14,   1;
  13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1;
  ...
		

References

• 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.

Crossrefs

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

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

Programs

Maple

egf:= k-> x^k / ((1-2*sinh(-x))*k!):
A341724:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341724(n,k), k=0..n)), n=0..8); # Mélika Tebni, Sep 04 2023

Formula

From Mélika Tebni, Sep 04 2023: (Start)

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

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

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

From Alois P. Heinz, Sep 04 2023: (Start)

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

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

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