A213831 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

1, 7, 4, 24, 19, 7, 58, 51, 31, 10, 115, 106, 78, 43, 13, 201, 190, 154, 105, 55, 16, 322, 309, 265, 202, 132, 67, 19, 484, 469, 417, 340, 250, 159, 79, 22, 693, 676, 616, 525, 415, 298, 186, 91, 25, 955, 936, 868, 763, 633

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213832.

Antidiagonal sums: A212560.

row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.

Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.

Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.

For a guide to related arrays, see A212500.

Examples

			1....7....24....58....115
4....19...51....106...190
7....31...78....154...265
10...43...105...202...340
13...55...132...250...415
		

Crossrefs

Cf. A212500

Programs

Mathematica

b[n_]:=2n-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}] (* A213831 *)
Table[t[n,n],{n,1,40}] (* A213832 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A212560 *)

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

Northwest corner (the array is read by falling antidiagonals):