A213566 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 5, 4, 15, 13, 9, 36, 33, 25, 16, 76, 71, 59, 41, 25, 148, 140, 120, 93, 61, 36, 273, 260, 228, 183, 135, 85, 49, 485, 464, 412, 340, 260, 185, 113, 64, 839, 805, 721, 604, 476, 351, 243, 145, 81, 1424, 1369, 1233, 1044, 836, 636, 456, 309, 181, 100

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213567.

Antidiagonal sums: A213570.

Row 1, (1,1,2,3,5,...)**(1,4,9,16,25,...): A053808.

Row 2, (1,1,2,3,5,...)**(4,9,16,25,...).

Row 3, (1,1,2,3,5,...)**(16,25,49,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....5....15....36....76
4....13...33....71....140
9....25...59....120...228
16...41...93....183...340
25...61...135...260...476
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened

Crossrefs

Cf. A213500, A213587.

Programs

GAP

F:=Fibonacci;; Flat(List([1..12], n-> List([1..n], k-> k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 ))); # G. C. Greubel, Jul 26 2019

Magma

F:=Fibonacci; [k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4: k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= n^2;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213566 *)
d = Table[t[n, n], {n, 1, 40}] (* A213567 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213570 *)
(* Second program *)
With[{F = Fibonacci}, Table[k*(k*F[n-k+3] +2*F[n-k+4]) + F[n-k+7] -(k+2) *(2*n-k+4) -(n-k+1)^2 -4, {n, 12}, {k, n}]//Flatten] (* G. C. Greubel, Jul 26 2019 *)

PARI

f=fibonacci;
for(n=1,12, for(k=1,n, print1(k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4, ", "))) \\ G. C. Greubel, Jul 26 2019

Sage

f=fibonacci; [[k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019

Formula

T(n,k) = 4*T(n,k-1)-5*T(n,k-2)+T(n,k-3)+2*T(n,k-4)-T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)^2 *x^2) and g(x) = (1 - x - x^2)*(1 - x )^3.

T(n,k) = n*(n*F(k+2) + 2*F(k+3)) + F(k+6) - (n+2)*(2*k+n+2) - k^2 - 4, F = A000045. - Ehren Metcalfe, Jul 10 2019