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A083861
Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.
4
0, 0, 1, 0, 1, 5, 0, 1, 5, 19, 0, 1, 5, 20, 65, 0, 1, 5, 21, 75, 211, 0, 1, 5, 22, 85, 275, 665, 0, 1, 5, 23, 95, 341, 1000, 2059, 0, 1, 5, 24, 105, 409, 1365, 3625, 6305, 0, 1, 5, 25, 115, 479, 1760, 5461, 13125, 19171, 0, 1, 5, 26, 125, 551, 2185, 7573, 21845, 47500, 58025
OFFSET
0,6
COMMENTS
Row n >= 0 of the array gives the solution to the recurrence b(k) = 5*b(k-1) + (n - 6)*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1. The rows are the binomial transforms of the rows of array A083857. The rows are the second binomial transforms of the generalized Fibonacci numbers in array A083856.
FORMULA
T(n, k) = (((5 + sqrt(4*n + 1))/2)^k - ((5 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1).
O.g.f. for row n >= 0: -x/(-1 + 5*x + (n-6)*x^2) . - R. J. Mathar, Dec 02 2007
From Petros Hadjicostas, Dec 25 2019: (Start)
T(n,k) = 5*T(n,k-1) + (n - 6)*T(n,k-2) for k >= 2 with T(n,0) = 0 and T(n,1) = 1 for all n >= 0.
T(n,k) = Sum_{i = 0..k} binomial(k,i) * A083857(n,i).
T(n,k) = Sum_{i = 0..k} Sum_{j = 0..i} binomial(k,i) * binomial(i,j) * A083856(n,j). (End)
EXAMPLE
Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
0, 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, ...
0, 1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, ...
0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, ...
0, 1, 5, 22, 95, 409, 1760, 7573, 32585, 140206, ...
0, 1, 5, 23, 105, 479, 2185, 9967, 45465, 207391, ...
0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, ...
0, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, ...
...
MAPLE
seq(seq(round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 27 2019
MATHEMATICA
T[n_, k_]:= Round[(((5 +Sqrt[4*n+1])/2)^k - ((5 -Sqrt[4*n+1])/2)^k)/Sqrt[4*n+1]]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 27 2019 *)
PROG
(PARI) T(n, k) = round( (((5+sqrt(4*n+1))/2)^k - ((5-sqrt(4*n+1))/2)^k)/sqrt(4*n + 1) );
for(n=0, 10, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Dec 27 2019
(Magma)
T:= func< n, k | Round( (((5+Sqrt(4*n+1))/2)^k - ((5-Sqrt(4*n+1))/2)^k)/Sqrt(4*n + 1) ) >;
[T(n-k, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 27 2019
(Sage) [[round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 27 2019
CROSSREFS
Rows include A001047 (n=0), A093131 (n=1), A002450 (n=2), A004254 (n=5), A000351 (n=6), A052918 (n=7), A015535 (n=8), A015536 (n=9), A015537 (n=10).
Cf. A083856 (second inverse binomial transform), A083856 (first inverse binomial transform), A082297 (main diagonal).
Sequence in context: A197515 A326185 A293508 * A097591 A318299 A164652
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, May 06 2003
EXTENSIONS
Name and various sections edited by Petros Hadjicostas, Dec 25 2019
STATUS
approved