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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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6
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
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REFERENCES
| 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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EXAMPLE
| 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
| T:= proc(n, k) (n+1)*(n+2)/2 * binomial(n, k); end;
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CROSSREFS
| Columns include A000217. Row sums are A001788. Cf. A094306.
Cf. A003506, A121547, A121306, A119800, A000217, A007318.
Sequence in context: A169944 A110952 A025250 * A057963 A112434 A050067
Adjacent sequences: A094302 A094303 A094304 * A094306 A094307 A094308
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KEYWORD
| nonn,tabl,easy
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 29 2004
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EXTENSIONS
| 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
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