A135065 A127733 * A007318 as infinite lower triangular matrices.

1, 4, 4, 9, 18, 9, 16, 48, 48, 16, 25, 100, 150, 100, 25, 36, 180, 360, 360, 180, 36, 49, 294, 735, 980, 735, 294, 49, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 81, 648, 2268, 4536, 5670, 4536, 2268, 648, 81, 100, 900, 3600, 8400, 12600, 12600, 8400, 3600

Offset: 0

Author

Gary W. Adamson, Nov 16 2007

Keywords

• nonn
• tabl
• easy

Comments

A135065 * [1/1, 1/2, 1/3, ...] = A066524: (1, 6, 21, 60, 155, ...).

Triangle T(n,k), 0 <= k <= n, read by rows, given by (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

Examples

			First few rows of the triangle:
   1;
   4,   4;
   9,  18,   9;
  16,  48,  48,  16;
  25, 100, 150, 100,  25;
  36, 180, 360, 360, 180,  36;
  49, 294, 735, 980, 735, 294,  49;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows
• Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

Crossrefs

Cf. A000290, A127733, A066524, A014477 (row sums), A084938.

Programs

Maple

with(combstruct):for n from 0 to 11 do seq(n*m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008

Mathematica

Flatten[Table[Binomial[n,k](n+1)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 12 2013 *)

Formula

T(n,k) = binomial(n,k)*(n+1)^2 = A007318(n,k)*A000290(n+1). - Philippe Deléham, Oct 27 2011

T(n-1,k-1) = Sum_{i=-k..k} (-1)^i*(k^2-i^2)*binomial(n,k+i)*binomial(n,k-i). - Mircea Merca, Apr 05 2012

G.f.: (-1 - x - x*y)/(x + x*y - 1)^3. - R. J. Mathar, Aug 12 2015

Extensions

Corrected by Zerinvary Lajos, Apr 09 2008