%I #34 Jul 11 2023 12:04:17

%S 1,4,4,9,18,9,16,48,48,16,25,100,150,100,25,36,180,360,360,180,36,49,

%T 294,735,980,735,294,49,64,448,1344,2240,2240,1344,448,64,81,648,2268,

%U 4536,5670,4536,2268,648,81,100,900,3600,8400,12600,12600,8400,3600

%N A127733 * A007318 as infinite lower triangular matrices.

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

%C 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

%H G. C. Greubel, <a href="/A135065/b135065.txt">Table of n, a(n) for the first 50 rows</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

%F T(n,k) = binomial(n,k)*(n+1)^2 = A007318(n,k)*A000290(n+1). - _Philippe Deléham_, Oct 27 2011

%F 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

%F G.f.: (-1 - x - x*y)/(x + x*y - 1)^3. - _R. J. Mathar_, Aug 12 2015

%e First few rows of the triangle:

%e 1;

%e 4, 4;

%e 9, 18, 9;

%e 16, 48, 48, 16;

%e 25, 100, 150, 100, 25;

%e 36, 180, 360, 360, 180, 36;

%e 49, 294, 735, 980, 735, 294, 49;

%p 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

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

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

%K nonn,tabl,easy

%O 0,2

%A _Gary W. Adamson_, Nov 16 2007

%E Corrected by _Zerinvary Lajos_, Apr 09 2008