A213821 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

2, 9, 4, 24, 16, 6, 50, 39, 23, 8, 90, 76, 54, 30, 10, 147, 130, 102, 69, 37, 12, 224, 204, 170, 128, 84, 44, 14, 324, 301, 261, 210, 154, 99, 51, 16, 450, 424, 378, 318, 250, 180, 114, 58, 18, 605, 576, 524, 455, 375, 290, 206

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A033431.

Antidiagonal sums: A176060.

Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002.

Row 2, (2,5,8,11,…)**(2,3,4,5,…): (k^3 + 5*k^2 + 2*k)/2.

Row 3, (1,2,3,4,…)**(8,11,14,17,…): (k^3 + 8*k^2 + 3*k)/2.

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
2….9….24…50….90
4….16…39…76…130
6….23…54…102…170
8….30…69…128…210
10…37…84…154…250
12…44…99…180…290
		

Crossrefs

Cf. A212500

Programs

Mathematica

b[n_]:=3n-1;c[n_]:=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}] (* A213821 *)
Table[t[n,n],{n,1,40}] (* A033431 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A176060 *)

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