A213747 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

1, 6, 3, 23, 16, 7, 72, 57, 36, 15, 201, 170, 125, 76, 31, 522, 459, 366, 261, 156, 63, 1291, 1164, 975, 758, 533, 316, 127, 3084, 2829, 2448, 2007, 1542, 1077, 636, 255, 7181, 6670, 5905, 5016, 4071, 3110, 2165, 1276, 511, 16398, 15375, 13842

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213748.

Antidiagonal sums: A213749.

Row 1, (1,3,7,15,31,...)**(1,3,7,15,31,...): A045618.

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

Row 3, (1,3,7,15,31,...)**(7,15,31,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6.....23....72.....201
3....16....57....170....459
7....36....125...366....975
15...76....261...758....1007
31...156...533...1542...4071
		

Links

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

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := -1 + 2^n; c[n_] := -1 + 2^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}]  (* A213747 *)
Table[t[n, n], {n, 1, 40}] (* A213748 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213749 *)

Formula

T(n,k) = 6*T(n,k-1)-13*T(n,k-2)+12*T(n,k-3)-4*T(n,k-4).

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