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A014410
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Elements in Pascal's triangle (by row) that are not 1.
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21
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2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 20, 15, 6, 7, 21, 35, 35, 21, 7, 8, 28, 56, 70, 56, 28, 8, 9, 36, 84, 126, 126, 84, 36, 9, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 13, 78
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OFFSET
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2,1
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COMMENTS
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Also, rows of triangle formed using Pascal's rule except begin and end n-th row with n+2. - Asher Auel.
Given the triangle signed by rows (+ - + ...) = M, with V = a variant of the Bernoulli numbers starting [1/2, 1/6, 0, -1/30, 0, 1/42, ...]; M*V = [1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
Also A014410 * [1/2, 1/6, 0, -1/30, 0, 1/42, 0, ...] = [1, 2, 3, 4, ...]. For an alternative way to derive the Bernoulli numbers from a modified version of Pascal's triangle see A135225. - Peter Bala, Dec 18 2014
This is Johannes Scheubel's (1494-1570) (also Scheybl, Schöblin) version of the arithmetical triangle from his 1545 book "De numeris et diversis rationibus". See the Kac reference, p. 396 and the Table 12.1 on p. 395.
The row sums give 2*A000225(n-1) = A000918(n) = 2*(2^n - 1), n >= 2. (See the second comment above).
The alternating row sums give repeat(2,0) = 2*A059841(n), n >= 2. (End)
T(n+1,k) is the number of k-facets of the n-simplex. - Jianing Song, Oct 22 2023
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REFERENCES
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Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396.
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LINKS
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FORMULA
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T(n,k) = binomial(n,k) = A007318(n,k), n >= 2, k = 1, 2, ..., n-1.
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EXAMPLE
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The triangle T(n,k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 11
2: 2
3: 3 3
4: 4 6 4
5: 5 10 10 5
6: 6 15 20 15 6
7: 7 21 35 35 21 7
8: 8 28 56 70 56 28 8
9: 9 36 84 126 126 84 36 9
10: 10 45 120 210 252 210 120 45 10
11: 11 55 165 330 462 462 330 165 55 11
12: 12 66 220 495 792 924 792 495 220 66 12
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MAPLE
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for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # Zerinvary Lajos, Dec 02 2007
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MATHEMATICA
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Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ]
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PROG
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(Haskell)
a014410 n k = a014410_tabl !! (n-2) !! (k-1)
a014410_row n = a014410_tabl !! (n-2)
a014410_tabl = map (init . tail) $ drop 2 a007318_tabl
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CROSSREFS
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A180986 is the same sequence but regarded as a square array.
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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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