A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.

1, -1, 1, 5, -2, 1, -31, 15, -3, 1, 257, -124, 30, -4, 1, -2671, 1285, -310, 50, -5, 1, 33305, -16026, 3855, -620, 75, -6, 1, -484471, 233135, -56091, 8995, -1085, 105, -7, 1, 8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 1

Offset: 0

Author

N. J. A. Sloane, Mar 04 2021

Keywords

• sign
• tabl

Comments

Conjectures from Mélika Tebni, Sep 09 2023: (Start)

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

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

Examples

			Triangle begins:
        1;
       -1,        1;
        5,       -2,      1;
      -31,       15,     -3,       1;
      257,     -124,     30,      -4,     1;
    -2671,     1285,   -310,      50,    -5,     1;
    33305,   -16026,   3855,    -620,    75,    -6,   1;
  -484471,   233135, -56091,    8995, -1085,   105,  -7,  1;
  8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 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 2.

Crossrefs

Column 0 is a signed version of A000556, column 1 is A341726.

Cf. A341724, A341725.

Programs

Maple

egf:= k-> exp(x)*x^k / ((1+2*sinh(x))*k!):
A341723:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023
second Maple program:
A341723:= (n, k)-> (-1)^(n-k)*binomial(n, k)*add(j!*combinat[fibonacci](j+1)*Stirling2(n-k,j), j=0.. n-k):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023

Formula

From Mélika Tebni, Sep 09 2023: (Start)

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

T(n,k) = (-1)^(n-k)*binomial(n,k)*A000556(n-k).

Recurrence: T(n,0) = (-1)^n*A000556(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)