A213828 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

2, 13, 5, 42, 28, 8, 98, 78, 43, 11, 190, 164, 114, 58, 14, 327, 295, 230, 150, 73, 17, 518, 480, 400, 296, 186, 88, 20, 772, 728, 633, 505, 362, 222, 103, 23, 1098, 1048, 938, 786, 610, 428, 258, 118, 26, 1505, 1449, 1324

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213829.

Antidiagonal sums: A213830.

Row 1, (1,4,7,10,...)**(2,5,8,11,...): (3*k^2 + k)/2.

Row 2, (1,4,7,10,...)**(5,8,11,14,...): (3*k^3 + 9*k^2 + 4*k)/2.

Row 3, (1,4,7,10,...)**(8,11,14,17,...): (3*k^3 + 18*k^2 + 7*k)/2.

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
2....13...42....98....190
5....28...78....164...295
8....43...114...230...400
11...58...150...296...505
14...73...186...362...610
17...88...222...428...715
		

Links

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

Crossrefs

Cf. A212500

Programs

Mathematica

b[n_]:=3n-2;c[n_]:=3n-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}] (* A213828 *)
d=Table[t[n,n],{n,1,40}] (* A213829 *)
d/2 (* A005915 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213830 *)

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*((6*n-4) - (3*n-8)*x - (3*n-5)*x^2) and g(x) = (1-x)^4.