A213751 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

1, 5, 2, 14, 9, 3, 30, 23, 13, 4, 55, 46, 32, 17, 5, 91, 80, 62, 41, 21, 6, 140, 127, 105, 78, 50, 25, 7, 204, 189, 163, 130, 94, 59, 29, 8, 285, 268, 238, 199, 155, 110, 68, 33, 9, 385, 366, 332, 287, 235, 180, 126, 77, 37, 10, 506, 485, 447, 396, 336, 271

Offset: 1

Author

Clark Kimberling, Jun 20 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A051662

Antidiagonal sums: A006325

row 1, (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330

row 2, (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986

row 3, (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6

row 4, (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...5....14...30....55....91
2...9....23...46....80....127
3...13...32...62....105...163
4...17...41...78....130...199
5...21...50...94....155...235
6...25...59...110...180...271
		

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := 2 n - 1; c[n_] := 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}]  (* A213751 *)
Table[t[n, n], {n, 1, 40}] (* A051662 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A006325 *)

Formula

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = n + x - (n - 1)*x^2 and g(x) = (1 - x )^4.