A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2*n - 3 + 2*h, n>=1, h>=1, and ** = convolution.

1, 6, 3, 21, 14, 5, 58, 43, 22, 7, 141, 110, 65, 30, 9, 318, 255, 162, 87, 38, 11, 685, 558, 369, 214, 109, 46, 13, 1434, 1179, 798, 483, 266, 131, 54, 15, 2949, 2438, 1673, 1038, 597, 318, 153, 62, 17, 5998, 4975, 3442, 2167, 1278, 711, 370, 175, 70

Offset: 1

Author

Clark Kimberling, Jun 20 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213757.

Antidiagonal sums: A213758.

Row 1, (1,3,7,15,31,...)**(1,3,5,7,9,...): A047520.

Row 2, (1,3,7,15,31,...)**(3,5,7,9,11,...).

Row 3, (1,3,7,15,31,...)**(5,7,9,11,13,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....21....58....141...318
3....14...43....110...255...558
5....22...65....162...369...798
7....30...87....214...483...1038
9....38...109...266...597...1278
11...46...131...318...711...1518
		

Links

• Clark Kimberling, Antidiagonals n = 1..40, flattened

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := -1 + 2^n; 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}]  (* A213756 *)
Table[t[n, n], {n, 1, 40}] (* A213757 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213758 *)

Formula

T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).

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