A083861 Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

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

Author

Paul Barry, May 06 2003

Keywords

• easy
• nonn
• tabl

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.

Examples

			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, ...
  ...
		

Links

• G. C. Greubel, Antidiagonals n = 0..100, flattened
• OEIS, Transformations of integer sequences.

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).

Programs

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

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 *)

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

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

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)

Extensions

Name and various sections edited by Petros Hadjicostas, Dec 25 2019