0,2

Sum of all possible sums of k numbers chosen from among the first n+1 numbers. Additive analogue of triangle of Stirling numbers of first kind (A008275).

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 anti-diagonals). - Thomas Wieder (thomas.wieder(AT)t-online.de), Aug 06 2006

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

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 DELEHAM (kolotoko(AT)wanadoo.fr), Oct 07 2007

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

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

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}.

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

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

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

Sequence in context: A049926 A110952 A025250 this_sequence A057963 A112434 A050067

Adjacent sequences: A094302 A094303 A094304 this_sequence A094306 A094307 A094308

nonn,tabl,easy

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 29 2004

Edited by Ralf Stephan, Feb 04 2005. Further comments from David Wasserman (dwasserm(AT)earthlink.net), Oct 04 2007

Further editing by N. J. A. Sloane (njas(AT)research.att.com), Oct 07 2007