A053201 -id:A053201 - cp's OEIS frontend

A053205 Row sums of A053201.

0, 0, 2, 0, 8, 0, 14, 6, 12, 0, 26, 0, 44, 36, 62, 0, 44, 0, 74, 48, 68, 0, 134, 30, 80, 78, 154, 0, 242, 0, 254, 72, 36, 86, 170, 0, 116, 84, 254, 0, 440, 0, 322, 330, 324, 0, 590, 126, 272, 414, 430, 0, 674, 96, 646, 120, 408, 0, 794, 0, 932, 384, 958, 290, 524, 0

Offset: 2

Asher Auel, Dec 12 1999

			a(6) = 0 + 3 + 2 + 3 + 0 = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 2..1000

Cf. A053201, A053202, A053204.

a053205 = sum . a053201_row  -- Reinhard Zumkeller, Jan 24 2014

a(n) = A053204(n) - 2.

a(prime(n)) = 0.

Corrected and extended by James Sellers, Dec 13 1999

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

Original entry on oeis.org

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

Mohammad K. Azarian

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

			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

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

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

Cf. A007318, A000918, A027641.
A180986 is the same sequence but regarded as a square array.
Cf. A000225,A059841, A257241 (Stifel's version).

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

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

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

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

More terms from Erich Friedman

A053200 Binomial coefficients C(n,k) reduced modulo n, read by rows; T(0,0)=0 by convention.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 2, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 0, 3, 0, 0, 3, 0, 0, 1, 1, 0, 5, 0, 0, 2, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 4, 3, 0, 0, 0, 3, 4, 6, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Asher Auel, Dec 12 1999

Pascal's triangle read by rows, where row n is read mod n.
A number n is a prime if and only if (1+x)^n == 1+x^n (mod n), i.e., if and only if the n-th row is 1,0,0,...,0,1. This result underlies the proof of Agrawal, Kayal and Saxena that there is a polynomial-time algorithm for primality testing. - N. J. A. Sloane, Feb 20 2004
A020475(n) = number of zeros in n-th row, for n > 0. - Reinhard Zumkeller, Jan 01 2013

			Row 4 = 1 mod 4, 4 mod 4, 6 mod 4, 4 mod 4, 1 mod 4 = 1, 0, 2, 0, 1.
Triangle begins:
  0;
  0,0;
  1,0,1;
  1,0,0,1;
  1,0,2,0,1;
  1,0,0,0,0,1;
  1,0,3,2,3,0,1;
  1,0,0,0,0,0,0,1;
  1,0,4,0,6,0,4,0,1;
  1,0,0,3,0,0,3,0,0,1;
  1,0,5,0,0,2,0,0,5,0,1;
  1,0,0,0,0,0,0,0,0,0,0,1;
  1,0,6,4,3,0,0,0,3,4,6,0,1;
  1,0,0,0,0,0,0,0,0,0,0,0,0,1;

T. D. Noe, Rows n = 0..100 of triangle, flattened
M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, Annals of Maths., 160:2 (2004), pp. 781-793. [alternate link]
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A053204. Cf. A053201, A053202, A053203, A007318 (Pascal's triangle).
Cf. also A092241.
Cf. A053214 (central terms, apart from initial 1).

a053200 n k = a053200_tabl !! n !! k
a053200_row n = a053200_tabl !! n
a053200_tabl = [0] : zipWith (map . flip mod) [1..] (tail a007318_tabl)
-- Reinhard Zumkeller, Jul 10 2015, Jan 01 2013

f := n -> seriestolist( series( expand( (1+x)^n ) mod n, x, n+1)); # N. J. A. Sloane

Flatten[Join[{0},Table[Mod[Binomial[n,Range[0,n]],n],{n,20}]]] (* Harvey P. Dale, Apr 29 2013 *)

T(n,k)=if(n, binomial(n,k)%n, 0) \\ Charles R Greathouse IV, Feb 07 2017

Corrected by T. D. Noe, Feb 08 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A053202 Pascal's triangle (excluding first, last two elements of each row) read by rows, row n read mod n.

Original entry on oeis.org

2, 0, 0, 3, 2, 3, 0, 0, 0, 0, 4, 0, 6, 0, 4, 0, 3, 0, 0, 3, 0, 5, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 3, 0, 0, 0, 3, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 7, 0, 7, 2, 7, 0, 7, 0, 7, 0, 5, 0, 3, 10, 0, 0, 10, 3, 0, 5, 0, 8, 0, 12, 0, 8, 0, 6, 0, 8, 0, 12, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 4

Asher Auel, Dec 12 1999

Prime numbered rows contain all zeros.

			Triangle begins:
  2;
  0, 0;
  3, 2, 3;
  0, 0, 0, 0;
  4, 0, 6, 0, 4;
  ...
row 8 = 28 mod 8, 56 mod 8, 70 mod 8, 56 mod 8, 28 mod 8 = 4, 0, 6, 0, 4.

T. D. Noe, Rows n = 4..100 of triangle, flattened

Sum of row n = A053205(n). Cf. A053200, A053201, A053203, A007318 (Pascal's triangle).

a053202 n k = a053202_tabl !! (n - 4) !! k
a053202_row n = a053202_tabl !! (n - 4)
a053202_tabl = zipWith (\k row -> take (k - 3) $ drop 2 row)
                       [4..] $ drop 4 a053200_tabl
-- Reinhard Zumkeller, Jan 24 2014

Table[Mod[Binomial[n, k], n], {n, 4, 18}, {k, 2, n-2}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

a(44) corrected by T. D. Noe, Feb 08 2008

A053203 Pascal's triangle (excluding first, last three elements of each row) read by rows, row n read mod n.

Original entry on oeis.org

2, 0, 0, 0, 6, 0, 3, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 7, 2, 7, 0, 7, 0, 5, 0, 3, 10, 0, 0, 10, 3, 0, 5, 0, 12, 0, 8, 0, 6, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 6, 0, 0, 2, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 6

Asher Auel, Dec 12 1999

Prime numbered rows contain all zeros.

			Triangle begins:
  2;
  0,0;
  0,6,0;
  3,0,0,3;
  0,0,2,0,0;
  ...
row 9 = 84 mod 9, 126 mod 9, 126 mod 9, 84 mod 9, = 3, 0, 0, 3.

T. D. Noe, Rows n=6..100 of triangle, flattened

Row sums give A053206.

Cf. A053200, A053201, A053203, A007318 (Pascal's triangle).

a053203 n k = a053203_tabl !! (n - 6) !! k
a053203_row n = a053203_tabl !! (n - 6)
a053203_tabl = zipWith (\k row -> take (k - 5) $ drop 3 row)
                       [6..] $ drop 6 a053200_tabl
-- Reinhard Zumkeller, Jan 24 2014

Table[Mod[Binomial[n, k], n], {n, 6, 20}, {k, 3, n-3}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)

a(30) corrected by T. D. Noe, Feb 08 2008

A053206 Row sums of A053203.

Original entry on oeis.org

2, 0, 6, 6, 2, 0, 14, 0, 30, 36, 46, 0, 26, 0, 54, 48, 46, 0, 110, 30, 54, 78, 126, 0, 212, 0, 222, 72, 2, 86, 134, 0, 78, 84, 214, 0, 398, 0, 278, 330, 278, 0, 542, 126, 222, 414, 378, 0, 620, 96, 590, 120, 350, 0, 734, 0, 870, 384, 894, 290, 458, 0, 150, 558, 742, 0, 1142

Offset: 6

Asher Auel, Dec 12 1999

			a(6) = 3 + 0 + 0 + 3 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 6..1000

Cf. A053201, A053202, A053204.

a053206 = sum . a053203_row  -- Reinhard Zumkeller, Jan 24 2014

a(n) = A053204(n) - n*((n+1) mod 2) - 2.

a(prime(n)) = 0, where prime(n) is the n-th prime.

Corrected and extended by James Sellers, Dec 13 1999

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A053205 Row sums of A053201.

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

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A053200 Binomial coefficients C(n,k) reduced modulo n, read by rows; T(0,0)=0 by convention.

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A053202 Pascal's triangle (excluding first, last two elements of each row) read by rows, row n read mod n.

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A053203 Pascal's triangle (excluding first, last three elements of each row) read by rows, row n read mod n.

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A053206 Row sums of A053203.

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