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A083856
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Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).
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8
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0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 5, 5, 1, 0, 1, 1, 5, 7, 11, 8, 1, 0, 1, 1, 6, 9, 19, 21, 13, 1, 0, 1, 1, 7, 11, 29, 40, 43, 21, 1, 0, 1, 1, 8, 13, 41, 65, 97, 85, 34, 1, 0, 1, 1, 9, 15, 55, 96, 181, 217, 171, 55, 1
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OFFSET
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0,14
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COMMENTS
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Row n >= 0 of the array gives the solution to the recurrence b(k) = b(k-1) + n*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1.
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LINKS
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FORMULA
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T(n, k) = (((1 + sqrt(4*n + 1))/2)^k - ((1 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1). [corrected by Michel Marcus, Jun 25 2018]
The g.f. for row n >= 0 is x/(1 - x - n*x^2).
The g.f. for column k >= 1 is g(k,x) = 1/(1-x) + Sum_{m = 1..floor((k-1)/2)} (1 - x)^(-1 - m) * binomial(k - 1 - m, m) * Sum_{i = 0..m} x^i * Sum_{j = 0..i} (-1)^j * (i - j)^m * binomial(1 + m, j).
The g.f. for column k >= 1 is also g(k,x) = 1 + Sum_{m = 1..floor((k+1)/2)} ((1 - x)^(-m) * binomial(k-m, m-1) * Sum_{j = 0..m} (-1)^j * binomial(m, j) * x^m * Phi(x, -m+1, -j+m)) + Sum_{s = 0..floor((k-1)/2)} binomial(k-s-1, s) * PolyLog(-s, x), where Phi is the Lerch transcendent function. (End)
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EXAMPLE
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Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A057427]
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... [A000045]
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, ... [A001045]
0, 1, 1, 4, 7, 19, 40, 97, 217, 508, ... [A006130]
0, 1, 1, 5, 9, 29, 65, 181, 441, 1165, ... [A006131]
0, 1, 1, 6, 11, 41, 96, 301, 781, 2286, ... [A015440]
0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, ... [A015441]
0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, ... [A015442]
0, 1, 1, 9, 17, 89, 225, 937, 2737, 10233, ... [A015443]
0, 1, 1, 10, 19, 109, 280, 1261, 3781, 15130, ... [A015445]
...
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MAPLE
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A083856_row := proc(r, n) local R; R := proc(n) option remember;
if n<=1 then n else R(n-1)+r*R(n-2) fi end: R(n) end:
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MATHEMATICA
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T[_, 0] = 0; T[_, 1|2] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
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PROG
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(Julia)
function generalized_fibonacci(r, n)
F = BigInt[1 r; 1 0]
Fn = F^n
Fn[2, 1]
end
for r in 0:6 println([generalized_fibonacci(r, n) for n in 0:9]) end # Peter Luschny, Mar 06 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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