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Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
10

%I #50 Jul 11 2023 12:04:21

%S 1,3,3,6,12,6,10,30,30,10,15,60,90,60,15,21,105,210,210,105,21,28,168,

%T 420,560,420,168,28,36,252,756,1260,1260,756,252,36,45,360,1260,2520,

%U 3150,2520,1260,360,45,55,495,1980,4620,6930,6930,4620,1980,495,55,66

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

%C 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 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 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 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 T(n,k) = A000217(n+1) * A007318(n,k), 0 <= k <= n. - _Reinhard Zumkeller_, Jul 30 2013

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

%H Reinhard Zumkeller, <a href="/A094305/b094305.txt">Rows n = 0..100 of table, flattened</a>

%H 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 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 O.g.f.: 1/(1 - x - y*x)^3. - _Geoffrey Critzer_, Apr 06 2013

%e Triangle begins:

%e 1

%e 3 3

%e 6 12 6

%e 10 30 30 10

%e 15 60 90 60 15

%e 21 105 210 210 105 21

%e ...

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

%t 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 (* _Geoffrey Critzer_, Apr 06 2013 *)

%t Flatten[Table[((n+1)(n+2))/2 Binomial[n,k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Aug 31 2014 *)

%o (Haskell)

%o a094305 n k = a094305_tabl !! n !! k

%o a094305_row n = a094305_tabl !! n

%o a094305_tabl = zipWith (map . (*)) (tail a000217_list) a007318_tabl

%o -- _Reinhard Zumkeller_, Jul 30 2013

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

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

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

%K nonn,tabl,easy

%O 0,2

%A _Amarnath Murthy_, Apr 29 2004

%E Edited by _Ralf Stephan_, Feb 04 2005

%E Further comments from _David Wasserman_, Oct 04 2007

%E Further editing by _N. J. A. Sloane_, Oct 07 2007