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A341724
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Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.
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6
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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
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OFFSET
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0,2
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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 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 3.
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LINKS
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FORMULA
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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)
|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)
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EXAMPLE
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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;
...
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MAPLE
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egf:= k-> x^k / ((1-2*sinh(-x))*k!):
A341724:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
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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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