A213753 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = -1 + 2^(n-1+h), n>=1, h>=1, and ** = convolution.

1, 6, 3, 21, 16, 7, 58, 51, 36, 15, 141, 132, 111, 76, 31, 318, 307, 280, 231, 156, 63, 685, 672, 639, 576, 471, 316, 127, 1434, 1419, 1380, 1303, 1168, 951, 636, 255, 2949, 2932, 2887, 2796, 2631, 2352, 1911, 1276, 511, 5998, 5979, 5928, 5823

Offset: 1

Author

Clark Kimberling, Jun 20 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213754.

Antidiagonal sums: A213755.

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

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

Row 3, (1,3,5,7,9,...)**(7,15,31,63...).

Ror a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6.....21....58.....141
3....16....51....132....307
7....36....111...280....639
15...76....231...576....1303
31...156...471...1168...2631
		

Links

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

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := 2 n - 1; 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}]  (* A213753 *)
Table[t[n, n], {n, 1, 40}] (* A213754 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213755 *)

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*(-1 + 2^n + x + (-2 + 2^n)*x^2) and g(x) = (1 - 2*x)(1 - x )^3.