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A014410
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Elements in Pascal's triangle (by row) that are not 1.
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19
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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 (asher.auel(AT)reed.edu).
Row sums are A000918. - Roger L. Bagula and Gary W. Adamson, Jan 15 2009
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
T(n,k) mod n = A053201(n,k), k=1..n-1. - Reinhard Zumkeller, Aug 17 2013
From Wolfdieter Lang, May 22 2015: (Start)
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)
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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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Reinhard Zumkeller, Rows n=2..150 of triangle, flattened
Carl McTague, On the Greatest Common Divisor of binomial(qn, q), binomial(qn,2q), ..., binomial(qn, qn-q), arXiv:1510.06696 [math.CO], 2015.
Wikipedia, Johannes Scheubel (in German).
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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.
a(n) = C(A003057(n),A002260(n)) = C(A003057(n),A004736(n)). - Lekraj Beedassy, Jul 29 2006
T(n,j) = ( Gamma[4 + n]/(Gamma[2 + j] Gamma[3 - j + n]) - KroneckerDelta[ -4 - n]). - Roger L. Bagula and Gary W. Adamson, Jan 15 2009
T(n,k) = A028263(n,k) - A007318(n,k). - Reinhard Zumkeller, Mar 12 2012
gcd{k=1..n-1} T(n, k) = A014963(n), see Theorem 1 of McTague link. - Michel Marcus, Oct 23 2015
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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
... reformatted. - Wolfdieter Lang, May 22 2015
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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
-- Reinhard Zumkeller, Mar 12 2012
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CROSSREFS
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Cf. A007318, A000918, A027641.
A180986 is the same sequence but regarded as a square array.
Cf. A000225,A059841, A257241 (Stifel's version).
Sequence in context: A203996 A059442 A225273 * A180986 A200763 A203291
Adjacent sequences: A014407 A014408 A014409 * A014411 A014412 A014413
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Mohammad K. Azarian
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EXTENSIONS
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More terms from Erich Friedman
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STATUS
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approved
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