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

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

Offset: 2

Author

Mohammad K. Azarian

Keywords

• nonn
• easy
• tabl

Comments

Also, rows of triangle formed using Pascal's rule except begin and end n-th row with n+2. - Asher Auel.

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)

T(n+1,k) is the number of k-facets of the n-simplex. - Jianing Song, Oct 22 2023

Examples

			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
		

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).
• Wikipedia, Simplex

Crossrefs

Cf. A007318, A000918, A027641.

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

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

Programs

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

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& ]

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

Extensions

More terms from Erich Friedman