A213773 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

1, 8, 4, 30, 23, 7, 76, 66, 38, 10, 155, 142, 102, 53, 13, 276, 260, 208, 138, 68, 16, 448, 429, 365, 274, 174, 83, 19, 680, 658, 582, 470, 340, 210, 98, 22, 981, 956, 868, 735, 575, 406, 246, 113, 25, 1360, 1332, 1232, 1078

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213782

Antidiagonal sums: A214092

Row 1, (1,4,7,10,…)**(1,4,7,10,…): A100175

Row 2, (1,4,7,10,…)**(4,7,10,13,…): (3*k^3 + 6*k^2 - k)/2

Row 3, (1,4,7,10,…)**(7,10,13,16,…): (3*k^3 + 15*k^2 - 4*k)/2

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....8....30....76....155...276
4....23...66....142...260...429
7....38...102...208...365...582
10...53...138...274...470...735
13...68...174...340...575...888
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1034

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_]:=3n-2;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}] (* A213773 *)
Table[t[n,n],{n,1,40}] (* A214092 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213818 *)

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-2 + (3*n+1)*x - (6*n-10)*x^2) and g(x) = (1-x)^4.