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A094305
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Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
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10
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 152.
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LINKS
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FORMULA
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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
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EXAMPLE
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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}.
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MAPLE
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A094305:= proc(n, k) (n+1)*(n+2)/2 * binomial(n, k); end;
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MATHEMATICA
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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
Flatten[Table[((n+1)(n+2))/2 Binomial[n, k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Aug 31 2014 *)
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PROG
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(Haskell)
a094305 n k = a094305_tabl !! n !! k
a094305_row n = a094305_tabl !! n
a094305_tabl = zipWith (map . (*)) (tail a000217_list) a007318_tabl
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CROSSREFS
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For a closely related array that also includes a row and column of zeros see A129533.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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