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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 (list; table; graph; refs; listen; history; text; internal format)
 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. LINKS G. C. Greubel, Antidiagonals n = 0..100, flattened 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 Adjacent sequences:  A083858 A083859 A083860 * A083862 A083863 A083864 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, May 06 2003 EXTENSIONS Name and various sections edited by Petros Hadjicostas, Dec 25 2019 STATUS approved

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Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)