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 A014410 Elements in Pascal's triangle (by row) that are not 1. 19

%I

%S 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,

%T 8,9,36,84,126,126,84,36,9,10,45,120,210,252,210,120,45,10,11,55,165,

%U 330,462,462,330,165,55,11,12,66,220,495,792,924,792,495,220,66,12,13,78

%N Elements in Pascal's triangle (by row) that are not 1.

%C 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).

%C Row sums are A000918. - _Roger L. Bagula_ and _Gary W. Adamson_, Jan 15 2009

%C 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

%C 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

%C T(n,k) mod n = A053201(n,k), k=1..n-1. - _Reinhard Zumkeller_, Aug 17 2013

%C From _Wolfdieter Lang_, May 22 2015: (Start)

%C This is Johannes Scheubel's (1494-1570) (also Scheybl, SchÃ¶blin) 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.

%C The row sums give 2*A000225(n-1) = A000918(n) = 2*(2^n - 1), n >= 2. (See the second comment above).

%C The alternating row sums give repeat(2,0) = 2*A059841(n), n >= 2.

%C (End)

%D Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396.

%H Reinhard Zumkeller, <a href="/A014410/b014410.txt">Rows n=2..150 of triangle, flattened</a>

%H Carl McTague, <a href="http://arxiv.org/abs/1510.06696">On the Greatest Common Divisor of binomial(qn, q), binomial(qn,2q), ..., binomial(qn, qn-q)</a>, arXiv:1510.06696 [math.CO], 2015.

%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Johann_Scheubel">Johannes Scheubel</a> (in German).

%F T(n,k) = binomial(n,k) = A007318(n,k), n >= 2, k = 1, 2, ...,n-1.

%F a(n) = C(A003057(n),A002260(n)) = C(A003057(n),A004736(n)). - _Lekraj Beedassy_, Jul 29 2006

%F 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

%F T(n,k) = A028263(n,k) - A007318(n,k). - _Reinhard Zumkeller_, Mar 12 2012

%F gcd{k=1..n-1} T(n, k) = A014963(n), see Theorem 1 of McTague link. - _Michel Marcus_, Oct 23 2015

%e The triangle T(n,k) begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 11

%e 2: 2

%e 3: 3 3

%e 4: 4 6 4

%e 5: 5 10 10 5

%e 6: 6 15 20 15 6

%e 7: 7 21 35 35 21 7

%e 8: 8 28 56 70 56 28 8

%e 9: 9 36 84 126 126 84 36 9

%e 10: 10 45 120 210 252 210 120 45 10

%e 11: 11 55 165 330 462 462 330 165 55 11

%e 12: 12 66 220 495 792 924 792 495 220 66 12

%e ... reformatted. - _Wolfdieter Lang_, May 22 2015

%p for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # _Zerinvary Lajos_, Dec 02 2007

%t Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ]

%o a014410 n k = a014410_tabl !! (n-2) !! (k-1)

%o a014410_row n = a014410_tabl !! (n-2)

%o a014410_tabl = map (init . tail) \$ drop 2 a007318_tabl

%o -- _Reinhard Zumkeller_, Mar 12 2012

%Y Cf. A007318, A000918, A027641.

%Y A180986 is the same sequence but regarded as a square array.

%Y Cf. A000225,A059841, A257241 (Stifel's version).

%K nonn,easy,tabl

%O 2,1