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A104712
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Pascal's triangle, with the first two columns removed.
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4
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1, 3, 1, 6, 4, 1, 10, 10, 5, 1, 15, 20, 15, 6, 1, 21, 35, 35, 21, 7, 1, 28, 56, 70, 56, 28, 8, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 78, 286, 715
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| A000295 (Eulerian numbers) gives the row sums.
Write A004736 and Pascal's triangle as infinite lower triangular matrices A and B; then A*B is this triangle.
Contribution by Peter Luschny, Apr 10 2011: (Start)
A slight variation has a combinatorial interpretation: remove the last column and the second one from Pascal's triangle. Let P(m, k) denote the set partitions of {1,2,..,n} with the following properties:
(a) Each partition has at least one singleton block;
(c) k is the size of the largest block of the partition;
(b) m = n - k + 1 is the number of parts of the partition.
Then A000295(n) = sum_{1 <= k <= n} card(P(n-k+1,k)).
For instance A000295(4) = P(4,1) + P(3,2) + P(2,3) + P(1,4) = card({1|2|3|4}) + card({1|2|34, 1|3|24,1|4|23, 2|3|14, 2|4|13, 3|4|12}) + card({1|234, 2|134, 3|124, 4|123}) = 1 + 6 + 4 = 11
This interpretation can be superimposed on the sequence by changing the offset to 1 and adding the value 1 in front. The triangle then starts
1
1,3
1,6,4
1,10,10,5
1,15,20,15,6 (End)
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FORMULA
| a(n, k) = binomial(n, k), for 2 <= k <= n.
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EXAMPLE
| Triangle begins
1
3 1
6 4 1
10 10 5 1
15 20 15 6 1
...
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MATHEMATICA
| t[n_, k_] := Binomial[n, k]; Table[ t[n, k], {n, 2, 13}, {k, 2, n}] // Flatten (*Robert G. Wilson v, Apr 16 2011 *)
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CROSSREFS
| Cf. A000295, A007318, A008292, A104713.
Sequence in context: A133110 A185915 A086270 * A122177 A108286 A185944
Adjacent sequences: A104709 A104710 A104711 * A104713 A104714 A104715
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KEYWORD
| nonn,tabl,easy,less
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2005
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EXTENSIONS
| Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), Jul 03 2007
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