A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 4, 1, 10, 5, 2, 21, 14, 9, 3, 40, 31, 24, 14, 5, 72, 61, 52, 38, 23, 8, 125, 112, 101, 83, 62, 37, 13, 212, 197, 184, 162, 135, 100, 60, 21, 354, 337, 322, 296, 263, 218, 162, 97, 34, 585, 566, 549, 519, 480, 425, 353, 262, 157, 55, 960, 939, 920, 886

Offset: 1

Author

Clark Kimberling, Jun 21 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213766.

Antidiagonal sums: A213767.

Row 1, (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.

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

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

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....4....10....21....40....72
1....5....14....31....61....112
2....9....24....52....101...184
3....14...38....83....162...296
5....23...62....135...263...480
8....37...100...218...425...776
13...60...162...353...688...1256
		

Links

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

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := 2 n - 1; c[n_] := Fibonacci[n];
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}]  (* A213765 *)
Table[t[n, n], {n, 1, 40}] (* A213766 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213767 *)

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*(F(n) + F(n+1)*x - F(n-1)*x^2) and g(x) = (1 - x - x^2)(1 - x )^2.

T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - Ehren Metcalfe, Jul 10 2019