A213762 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

1, 5, 3, 15, 11, 5, 37, 29, 17, 7, 83, 67, 43, 23, 9, 177, 145, 97, 57, 29, 11, 367, 303, 207, 127, 71, 35, 13, 749, 621, 429, 269, 157, 85, 41, 15, 1515, 1259, 875, 555, 331, 187, 99, 47, 17, 3049, 2537, 1769, 1129, 681, 393, 217, 113, 53, 19, 6119

Offset: 1

Author

Clark Kimberling, Jun 20 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213763.

Antidiagonal sums: A213764.

Row 1, (1,2,4,8,16,...)**(1,3,5,7,9,...): A050488.

Row 2, (1,2,4,8,16,...)**(3,5,7,9,11,...).

Row 3, (1,2,4,8,16,...)**(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....15...37....83....177
3....11...29...67....145...303
5....17...43...97....207...429
7....23...57...127...269...555
9....29...71...157...331...681
11...35...85...187...393...807
		

Links

• Clark Kimberling, Antidiagonals n = 1..80, flattened

Crossrefs

Cf. A213500, A213756.

Programs

Mathematica

b[n_] := 2^(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}]  (* A213762 *)
Table[t[n, n], {n, 1, 40}] (* A213763 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213764 *)

Formula

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

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 )^2.