A213774 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 5, 3, 14, 11, 5, 31, 26, 17, 7, 61, 53, 38, 23, 9, 112, 99, 75, 50, 29, 11, 197, 176, 137, 97, 62, 35, 13, 337, 303, 240, 175, 119, 74, 41, 15, 566, 511, 409, 304, 213, 141, 86, 47, 17, 939, 850, 685, 515, 368, 251, 163, 98, 53, 19, 1545, 1401, 1134

Offset: 1

Author

Clark Kimberling, Jun 21 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213775.

Antidiagonal sums: A213776.

Row 1, (1,2,3,5,8,...)**(1,3,5,7,9,...): A023652.

Row 2, (1,2,3,5,8,...)**(3,5,7,9,11,...).

Row 3, (1,2,3,5,8,...)**(5,7,9,11,13,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...31....61....112
3....11...26...53....99....176
5....17...38...75....137...240
7....23...50...97....175...304
9....29...62...119...213...368
11...35...74...141...251...432
		

Links

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

Crossrefs

Cf. A023652, A213500, A213768, A213775, A213776.

Programs

Mathematica

b[n_] := Fibonacci[n + 1]; c[n_] := 2 n - 1;
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}]  (* A213774 *)
Table[t[n, n], {n, 1, 40}] (* A213775 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213776 *)

Formula

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

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

T(n,k) = 2*n*Fibonacci(k+3) + Lucas(k+3) - 4*(k+n+1). - Ehren Metcalfe, Jul 08 2019