A213555 Rectangular array: (row n) = b**c, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

1, 10, 2, 46, 19, 3, 146, 82, 28, 4, 371, 246, 118, 37, 5, 812, 596, 346, 154, 46, 6, 1596, 1253, 821, 446, 190, 55, 7, 2892, 2380, 1694, 1046, 546, 226, 64, 8, 4917, 4188, 3164, 2135, 1271, 646, 262, 73, 9, 7942, 6942, 5484, 3948, 2576, 1496, 746

Offset: 1

Author

Clark Kimberling, Jun 17 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213556.

Antidiagonal sums: A213547.

Row 1, (1,8,27,...)**(1,2,3,...): A024166.

Row 2, (1,8,27,...)**(2,3,4,...): (3*k^5 + 30*k^4 + 55*k^3 + 30*k^2 + 2*k)/60.

Row 3, (1,8,27,...)**(3,4,5,...): (3*k^5 + 45*k^4 + 85*k^3 + 45*k^2 + 2*k)/60.

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...10...46....146...371....812
2...19...82....246...596....1253
3...28...118...346...821....1694
4...37...154...446...1046...2135
5...46...190...546...1271...2576
6...55...226...646...1496...3017
		

Links

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

Crossrefs

Cf. A213500, A213553.

Programs

Mathematica

b[n_] := n^3; 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}]  (* A213555 *)
d = Table[t[n, n], {n, 1, 40}] (* A213556 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

Formula

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) -T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n + (3*n + 1)*x - (3*n - 4)*x^2 - (n - 1)*x^3 and g(x) = (1 - x)^6.