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A341723
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Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.
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5
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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
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OFFSET
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0,4
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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FORMULA
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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)
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EXAMPLE
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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;
...
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MAPLE
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egf:= k-> exp(x)*x^k / ((1+2*sinh(x))*k!):
A341723:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
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):
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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