%I A094305
%S A094305 1,3,3,6,12,6,10,30,30,10,15,60,90,60,15,21,105,210,210,105,21,28,168,
%T A094305 420,560,420,168,28,36,252,756,1260,1260,756,252,36,45,360,1260,2520,
%U A094305 3150,2520,1260,360,45,55,495,1980,4620,6930,6930,4620,1980,495,55,66
%N A094305 Triangle read by rows: T(n,k) = (n+1)(n+2)/2 * binomial(n,k) (0 <= k 
               <= n).
%C A094305 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).
%C A094305 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
%C A094305 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
%C A094305 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
%D A094305 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, identity 152.
%H A094305 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/">
               Home Page</a>.
%H A094305 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder">(Old) 
               Home Page</a>.
%e A094305 Triangle begins:
%e A094305 1
%e A094305 3 3
%e A094305 6 12 6
%e A094305 10 30 30 10
%e A094305 15 60 90 60 15
%e A094305 21 105 210 210 105 21
%e A094305 ...
%e A094305 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}.
%p A094305 T:= proc(n,k) (n+1)*(n+2)/2 * binomial(n,k); end;
%Y A094305 Columns include A000217. Row sums are A001788. Cf. A094306.
%Y A094305 Cf. A003506, A121547, A121306, A119800, A000217, A007318.
%Y A094305 Sequence in context: A049926 A110952 A025250 this_sequence A057963 A112434 
               A050067
%Y A094305 Adjacent sequences: A094302 A094303 A094304 this_sequence A094306 A094307 
               A094308
%K A094305 nonn,tabl,easy
%O A094305 0,2
%A A094305 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 29 2004
%E A094305 Edited by Ralf Stephan, Feb 04 2005. Further comments from David Wasserman 
               (dwasserm(AT)earthlink.net), Oct 04 2007
%E A094305 Further editing by N. J. A. Sloane (njas(AT)research.att.com), Oct 07 
               2007