A094305 Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).

1, 3, 3, 6, 12, 6, 10, 30, 30, 10, 15, 60, 90, 60, 15, 21, 105, 210, 210, 105, 21, 28, 168, 420, 560, 420, 168, 28, 36, 252, 756, 1260, 1260, 756, 252, 36, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55, 66

Offset: 0

Author

Amarnath Murthy, Apr 29 2004

Keywords

• nonn
• tabl
• easy

Comments

Sum of all possible sums of k+1 numbers chosen from among the first n+1 numbers. Additive analog of triangle of Stirling numbers of first kind (A008275). - David Wasserman, Oct 04 2007

Third slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o)+a(m,n-1,o)+a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for which the first slice is Pascal's triangle (slice read by antidiagonals). - Thomas Wieder, Aug 06 2006

Triangle T(n,k), 0<=k<=n, read by rows given by [3,-1,2/3,-1/6,1/2,0,0,0,0,0,0,...] DELTA [3,-1,2/3,-1/6,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

T(n,k) is the number of ordered triples of bit strings with n bits and exactly k 1's over all bits in the triple. For example for n=1 we have (0,e,e),(e,0,e),(e,e,0),(1,e,e),(e,1,e),(e,e,1) where e is the empty string. - Geoffrey Critzer, Apr 06 2013

T(n,k) = A000217(n+1) * A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Jul 30 2013

Examples

			Triangle begins:
  1
  3 3
  6 12 6
  10 30 30 10
  15 60 90 60 15
  21 105 210 210 105 21
  ...
The n-th row is the product of the n-th triangular number and the n-th row of Pascal's triangle. The fifth row is (15,60,90,60,15) or 15*{1,4,6,4,1}.
		

References

• A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 152.

Links

• Reinhard Zumkeller, Rows n = 0..100 of table, flattened
• Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

Crossrefs

For a closely related array that also includes a row and column of zeros see A129533.

Columns include A000217. Row sums are A001788. Cf. A094306.

Cf. A003506, A121547, A121306, A119800, A000217, A007318.

Programs

Haskell

a094305 n k = a094305_tabl !! n !! k
a094305_row n = a094305_tabl !! n
a094305_tabl = zipWith (map . (*)) (tail a000217_list) a007318_tabl
-- Reinhard Zumkeller, Jul 30 2013

Maple

A094305:= proc(n,k) (n+1)*(n+2)/2 * binomial(n,k); end;

Mathematica

nn=10; f[list_]:=Select[list,#>0&];a=1/(1-x-y x); Map[f,CoefficientList[Series[a^3,{x,0,nn}],{x,y}]]//Grid
(* Geoffrey Critzer, Apr 06 2013 *)
Flatten[Table[((n+1)(n+2))/2 Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 31 2014 *)

Formula

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

O.g.f.: 1/(1 - x - y*x)^3. - Geoffrey Critzer, Apr 06 2013

Extensions

Edited by Ralf Stephan, Feb 04 2005

Further comments from David Wasserman, Oct 04 2007

Further editing by N. J. A. Sloane, Oct 07 2007