login
A135065
A127733 * A007318 as infinite lower triangular matrices.
1
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
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
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Nov 16 2007
EXTENSIONS
Corrected by Zerinvary Lajos, Apr 09 2008
STATUS
approved