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 A014410 Elements in Pascal's triangle (by row) that are not 1. 20
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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, 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. 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) REFERENCES Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396. LINKS 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). FORMULA 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,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 EXAMPLE 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 MAPLE for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # Zerinvary Lajos, Dec 02 2007 MATHEMATICA Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ] PROG (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 CROSSREFS 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 KEYWORD nonn,easy,tabl AUTHOR EXTENSIONS More terms from Erich Friedman STATUS approved

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Last modified April 20 01:34 EDT 2021. Contains 343117 sequences. (Running on oeis4.)