A213849 Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.

1, 2, 1, 5, 3, 2, 8, 6, 4, 2, 14, 11, 9, 5, 3, 20, 17, 14, 10, 6, 3, 30, 26, 23, 17, 13, 7, 4, 40, 36, 32, 26, 20, 14, 8, 4, 55, 50, 46, 38, 32, 23, 17, 9, 5, 70, 65, 60, 52, 44, 35, 26, 18, 10, 5, 91, 85, 80, 70, 62, 50, 41, 29, 21, 11, 6

Offset: 1

Author

Clark Kimberling, Jul 05 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A049778.

Antidiagonal sums: A213850.

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

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

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

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80
		

Links

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

Crossrefs

Cf. A212500.

Programs

Mathematica

b[n_]:=Floor[(n+1)/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}]  (* A213849 *)
d=Table[t[n,n],{n,1,50}] (* A049778 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213850 *)

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) = x*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.