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A341723
Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.
5
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
0,4
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)
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.
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)
EXAMPLE
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;
...
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
CROSSREFS
Column 0 is a signed version of A000556, column 1 is A341726.
Sequence in context: A300051 A281890 A342381 * A111544 A109281 A133289
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Mar 04 2021
STATUS
approved