A213825 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

2, 13, 8, 42, 34, 14, 98, 87, 55, 20, 190, 176, 132, 76, 26, 327, 310, 254, 177, 97, 32, 518, 498, 430, 332, 222, 118, 38, 772, 749, 669, 550, 410, 267, 139, 44, 1098, 1072, 980, 840, 670, 488, 312, 160, 50, 1505, 1476, 1372

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213826

Antidiagonal sums: A213827

Row 1, (2,5,8,13,...)**(1,4,7,10,13,...): (3*k^2 + k)/2

Row 2, (2,5,8,13,...)**(4,7,10,13,...): (3*k^3 + 9*k^2 - 2*k)/2

Row 3, (2,5,8,13,...)**(7,10,13,16,...): (3*k^3 + 18*k^2 - 5*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
8....34....87....176...310
14...55....132...254...430
20...76....177...332...550
26...97....222...410...670
32...118...267...488...790
		

Links

• Clark Kimberling, Antidiagonals n = 1..80, flattened

Crossrefs

Cf. A212500

Programs

Mathematica

b[n_]:=3n-1;c[n_]:=3n-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}] (* A213825 *)
d=Table[t[n,n],{n,1,40}] (* A213826 *)
d/2 (* A024215 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213827 *)

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