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

%I A094305 #50 Jul 11 2023 12:04:21
%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+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
%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 antidiagonals). - _Thomas Wieder_, Aug 06 2006
%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 Deléham_, Oct 07 2007
%C A094305 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
%C A094305 T(n,k) = A000217(n+1) * A007318(n,k), 0 <= k <= n. - _Reinhard Zumkeller_, Jul 30 2013
%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 Reinhard Zumkeller, <a href="/A094305/b094305.txt">Rows n = 0..100 of table, flattened</a>
%H A094305 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 A094305 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
%F A094305 O.g.f.: 1/(1 - x - y*x)^3.  - _Geoffrey Critzer_, Apr 06 2013
%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 A094305:= proc(n,k) (n+1)*(n+2)/2 * binomial(n,k); end;
%t A094305 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
%t A094305 (* _Geoffrey Critzer_, Apr 06 2013 *)
%t A094305 Flatten[Table[((n+1)(n+2))/2 Binomial[n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Aug 31 2014 *)
%o A094305 (Haskell)
%o A094305 a094305 n k = a094305_tabl !! n !! k
%o A094305 a094305_row n = a094305_tabl !! n
%o A094305 a094305_tabl = zipWith (map . (*)) (tail a000217_list) a007318_tabl
%o A094305 -- _Reinhard Zumkeller_, Jul 30 2013
%Y A094305 For a closely related array that also includes a row and column of zeros see A129533.
%Y A094305 Columns include A000217. Row sums are A001788. Cf. A094306.
%Y A094305 Cf. A003506, A121547, A121306, A119800, A000217, A007318.
%K A094305 nonn,tabl,easy
%O A094305 0,2
%A A094305 _Amarnath Murthy_, Apr 29 2004
%E A094305 Edited by _Ralf Stephan_, Feb 04 2005
%E A094305 Further comments from _David Wasserman_, Oct 04 2007
%E A094305 Further editing by _N. J. A. Sloane_, Oct 07 2007