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

0,2

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

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.

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

Example

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;

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 *)

Crossrefs

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

Sequence in context: A339427 A319646 A214826 * A067553 A112683 A192030

Adjacent sequences: A135062 A135063 A135064 * A135066 A135067 A135068

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Nov 16 2007

Extensions

Corrected by Zerinvary Lajos, Apr 09 2008

Status

approved