A213844 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-5+4*h, n>=1, h>=1, and ** = convolution.

3, 16, 7, 47, 32, 11, 104, 83, 48, 15, 195, 168, 119, 64, 19, 328, 295, 232, 155, 80, 23, 511, 472, 395, 296, 191, 96, 27, 752, 707, 616, 495, 360, 227, 112, 31, 1059, 1008, 903, 760, 595, 424, 263, 128, 35, 1440, 1383, 1264

Offset: 1

Author

Clark Kimberling, Jul 05 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213845.

Antidiagonal sums: A213846.

Row 1, (1,3,5,7...)**(3,7,11,15,...): A172482.

Row 2, (1,3,5,7,...)**(7,11,15,19,...): (4*k^3 + 15*k^2 + 2*k)/3.

Row 3, (1,3,5,7,...)**(11,15,19,23,...): (4*k^3 + 27*k^2 + 2*k)/3.

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
3....16...47....104...195...328
7....32...83....168...295...472
11...48...119...232...395...616
15...64...155...296...495...760
		

Links

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

Crossrefs

Cf. A212500.

Programs

Mathematica

b[n_]:=2n-1;c[n_]:=4n-1;
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}] (* A213844 *)
Table[t[n,n],{n,1,40}] (* A213845 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213846 *)

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-1 + 4*x - (4*n-5)*x^2) and g(x) = (1-x)^4.