A213783 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

1, 3, 1, 6, 4, 2, 11, 8, 6, 2, 17, 14, 11, 7, 3, 26, 22, 19, 13, 9, 3, 36, 32, 28, 22, 16, 10, 4, 50, 45, 41, 33, 27, 18, 12, 4, 65, 60, 55, 47, 39, 30, 21, 13, 5, 85, 79, 74, 64, 56, 44, 35, 23, 15, 5, 106, 100, 94, 84, 74, 62, 50, 38, 26, 16, 6, 133, 126, 120, 108

Offset: 1

Author

Clark Kimberling, Jun 22 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213759.

Antidiagonal sums: A213760.

Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744.

Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...).

Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...3....6....11...17...26...36....50
1...4....8....14...22...32...45....60
2...6....11...19...28...41...55....74
2...7....13...22...33...47...64....84
3...9....16...27...39...56...74....98
3...10...18...30...44...62...83....108
4...12...21...35...50...71...93....122
4...13...23...38...55...77...102...132
		

Links

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

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
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}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213760 *)

Formula

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

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