A213835 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

1, 7, 5, 22, 19, 9, 50, 46, 31, 13, 95, 90, 70, 43, 17, 161, 155, 130, 94, 55, 21, 252, 245, 215, 170, 118, 67, 25, 372, 364, 329, 275, 210, 142, 79, 29, 525, 516, 476, 413, 335, 250, 166, 91, 33, 715, 705, 660, 588, 497, 395

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A172078.

Antidiagonal sums: A051797.

Row 1, (1,2,3,4,5,...)**(1,5,9,13,...): A002412.

Row 2, (1,2,3,4,5,...)**(5,9,13,17,...): (4*k^3 + 15*k^2 - 11*k)/6.

Row 3, (1,2,3,4,5,...)**(9,13,17,21,...): (4*k^3 + 27*k^2 - 23*k)/6

For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....7....22....50....95
5....19...46....90....155
9....31...70....130...215
13...43...94....170...275
17...55...118...210...335
21...67...142...250...395
		

Links

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

Crossrefs

Cf. A212500.

Cf. A304659 (first lower diagonal).

Programs

Mathematica

b[n_]:=n;c[n_]:=4n-3;
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}] (* A213835 *)
Table[t[n,n],{n,1,40}] (* A172078 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A051797 *)

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*((4*n-3) + (4*n-7)*x) and g(x) = (1-x)^4.