A213841 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

1, 8, 5, 29, 24, 9, 72, 65, 40, 13, 145, 136, 101, 56, 17, 256, 245, 200, 137, 72, 21, 413, 400, 345, 264, 173, 88, 25, 624, 609, 544, 445, 328, 209, 104, 29, 897, 880, 805, 688, 545, 392, 245, 120, 33, 1240, 1221, 1136, 1001

Offset: 1

Author

Clark Kimberling, Jul 05 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213842.

Antidiagonal sums: A213843.

Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.

Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 + 2*k)/3.

Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 + 2*k)/3.

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....8....29....72....145
5....24...65....136...245
9....40...101...200...345
13...56...137...264...445
17...72...173...328...545
21...88...209...392...645
		

Links

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

Crossrefs

Cf. A212500.

Programs

Mathematica

b[n_]:=2n-1;c[n_]:=4n-3;
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}] (* A213841 *)
Table[t[n,n],{n,1,40}] (* A213842 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213843 *)

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*(4*n-3 + 4*x - (4*n-7)*x^2) and g(x) = (1-x)^4.