A053200 -id:A053200 - cp's OEIS frontend

A053204 Row sums of A053200.

0, 0, 2, 2, 4, 2, 10, 2, 16, 8, 14, 2, 28, 2, 46, 38, 64, 2, 46, 2, 76, 50, 70, 2, 136, 32, 82, 80, 156, 2, 244, 2, 256, 74, 38, 88, 172, 2, 118, 86, 256, 2, 442, 2, 324, 332, 326, 2, 592, 128, 274, 416, 432, 2, 676, 98, 648, 122, 410, 2, 796, 2, 934, 386, 960, 292, 526

Offset: 0

Asher Auel, Dec 12 1999

a(prime(n)) = 2, where prime(n) = prime numbers A000040, excluding the case where a(1) = 2.

			a(6) = 1 + 0 + 3 + 2 + 3 + 0 + 1 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A053200.

a053204 = sum . a053200_row  -- Reinhard Zumkeller, Jan 01 2013

Corrected and extended by James Sellers, Dec 13 1999

Initial terms adjusted by Reinhard Zumkeller, Jan 01 2013

A020475 a(n) is the number of k for which binomial(n,k) is divisible by n.

Original entry on oeis.org

0, 2, 1, 2, 2, 4, 2, 6, 4, 6, 6, 10, 5, 12, 6, 8, 8, 16, 10, 18, 10, 14, 14, 22, 10, 20, 18, 18, 14, 28, 11, 30, 16, 26, 30, 26, 22, 36, 30, 30, 22, 40, 20, 42, 26, 26, 30, 46, 20, 42, 34, 32, 34, 52, 26, 46, 33, 50, 42, 58, 26, 60, 30, 46, 32, 50, 48, 66, 58, 50, 44, 70, 40, 72, 66, 46, 58

Offset: 0

David W. Wilson

nonn

Note that n is prime iff a(n) = n-1. - T. D. Noe, Feb 23 2006
a(n) >= phi(n) (cf. Robbins). - Michel Marcus, Oct 31 2012
For n > 0: number of zeros in n-th row of A053200. - Reinhard Zumkeller, Jan 01 2013
binomial(n, k) / binomial(n, k-1) = k / (n - k + 1) which eases computation. - David A. Corneth, May 04 2025

			From _David A. Corneth_, May 04 2025: (Start)
a(6) = 2 as binomial(6, k) = 1, 6, 15, 20, 15, 6, 1 for k = 0 through 6 respectively. Among these, a(6) = 2 numbers are divisible by 6.
a(12) = 5. We start at 1 and know that 12 = 2^2 * 3^1 and so the valuation of 2 and 3 are 2 and 1 respectively.
We compute binomial(n, k) / binomial(n, k-1) = k / (n - k + 1) for k = 1 through floor((n-1)/2) = 3 and keep track the valuation of 2 and 3 of binomial(n, k) via this telescoping product.
We get (2, 1), (1, 1), (2, 0), (0, 2), (3, 2). Of them (2, 1) and (3, 2) indicate divisibility by 12 so that's two numbers and four via symmetry. For binomial(12, 6) we get (2, 1) which as symmetry doesn't give two solutions only counts towards one value of k. This gives a total 4+1 = 5 for a(12). (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
H. Harborth, Divisibility of binomial coefficients by their row number, The American Mathematical Monthly, Vol. 84, No. 1 (Jan., 1977), pp. 35-37.
N. Robbins, On the number of binomial coefficients which are divisible by their row number, Canad. Math. Bull. 25(1982), 363-365.
N. Robbins, On the number of binomial coefficients which are divisible by their row number. II, Canad. Math. Bull. 28(1985), 481-486.
David A. Corneth, PARI program

Cf. A007012.

a020475 n = a020475_list !! n
a020475_list = 0 : map (sum . map (0 ^)) (tail a053200_tabl)
-- Reinhard Zumkeller, Jan 24 2014

Table[cnt=0; Do[If[Mod[Binomial[n,k],n]==0, cnt++ ], {k,0,n}]; cnt,{n,0,100}] (* T. D. Noe, Feb 23 2006 *)
Join[{0},Table[Count[Table[Binomial[n,k],{k,0,n}],?(Mod[#,n]==0&)],{n,100}]] (* _Harvey P. Dale, Sep 20 2024 *)

apply( A020475(n)=sum(k=0,n, binomial(n,k)%n==0), [1..30]) \\ M. F. Hasler, May 04 2025

PARI
```
\\ See Corneth link
```

a(n) = n + 1 - A007012(n). - T. D. Noe, Feb 23 2006

More terms from T. D. Noe, Feb 23 2006

A053201 Pascal's triangle (excluding first, last element of each row) read by rows, row n read mod n.

Original entry on oeis.org

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

Offset: 2

Asher Auel, Dec 12 1999

Prime numbered rows contain all zeros.

			0; 0,0; 0,2,0; 0,0,0,0; 0,3,2,3,0; ...
row 6 = 6 mod 6, 15 mod 6, 20 mod 6, 15 mod 6, 6 mod 6 = 0, 3, 2, 3, 0

T. D. Noe, Rows n=2..100 of triangle, flattened

Row sums give A053205. Cf. A053200, A053202, A053203, A007318 (Pascal's triangle)

Cf. A053214 (central terms).

a053201 n k = a053201_tabl !! (n-2) !! (k-1)
a053201_row n = a053201_tabl !! (n-2)
a053201_tabl = zipWith (map . (flip mod)) [2..] a014410_tabl
-- Reinhard Zumkeller, Aug 17 2013

row[n_] := Table[ Mod[ Binomial[n, k], n], {k, 1, n-1}]; Table[row[n], {n, 2, 15}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)

T(n,k) = A014410(n,k) mod n, k=1..n-1.

a(62) and a(68) corrected by T. D. Noe, Feb 08 2008

A053214 Central binomial coefficients (A000984) read mod 2n, with a(0)=1.

Original entry on oeis.org

1, 0, 2, 2, 6, 2, 0, 2, 6, 2, 16, 2, 4, 2, 20, 0, 6, 2, 24, 2, 20, 6, 28, 2, 12, 2, 32, 20, 0, 2, 4, 2, 6, 42, 40, 42, 52, 2, 44, 20, 20, 2, 0, 2, 48, 0, 52, 2, 60, 2, 56, 54, 96, 2, 60, 32, 88, 96, 64, 2, 96, 2, 68, 12, 70, 70, 0, 2, 36, 66, 40, 2, 36, 2, 80, 120, 32, 0, 144, 2, 20, 20, 88

Offset: 0

Asher Auel, Dec 16 1999

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A059288, A059289.

a053214 0 = 1
a053214 n = a053200 (2 * n) n  -- Reinhard Zumkeller, Jan 24 2014

Join[{1}, Table[Mod[Binomial[2*n, n], 2*n], {n, 1, 100}]] (* G. C. Greubel, Sep 04 2018 *)

concat([1], vector(100, n, lift(Mod(binomial(2*n,n), 2*n)))) \\ G. C. Greubel, Sep 04 2018

a(n) = binomial(2*n, n) mod 2*n, with a(0)=1.

a(n) = A053200(2*n,n) for n > 0. - Reinhard Zumkeller, Jan 01 2013

More terms from James Sellers, Dec 18 1999

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

A050870 T(h,k) = binomial(h,k) - A050186(h,k).

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, 10, 2, 10, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 4, 15, 0, 24, 0, 15, 4, 6, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 7, 0, 21, 0, 35, 2, 35, 0, 21, 0, 7, 0, 1, 1, 0

Offset: 0

Clark Kimberling

T(h,k) = number of periodic binary words of k 1's and h-k 0's.

			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;
1,0,5,0,10,2,10,0,5,0,1;

Cf. A007318. Different from A053200.

A050186 := proc(n,k)
        if n = 0 then
                1;
        else
        add (numtheory[mobius](d)*binomial(n/d,k/d),d =numtheory[divisors](igcd(n,k))) ;
        end if;
end proc:
A050870 := proc(n,k)
        binomial(n,k)-A050186(n,k) ;
end proc:
seq(seq(A050870(n,k),k=0..n),n=0..20) ; # R. J. Mathar, Sep 24 2011

T[n_, k_] := Binomial[n, k] - If[n == 0, 1, Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* Jean-François Alcover, Jul 01 2019 *)

Edited by N. J. A. Sloane, Aug 29 2008

A007012 a(n) is number of k for which C(n,k) is not divisible by n.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 5, 2, 8, 2, 9, 8, 9, 2, 9, 2, 11, 8, 9, 2, 15, 6, 9, 10, 15, 2, 20, 2, 17, 8, 5, 10, 15, 2, 9, 10, 19, 2, 23, 2, 19, 20, 17, 2, 29, 8, 17, 20, 19, 2, 29, 10, 24, 8, 17, 2, 35, 2, 33, 18, 33, 16, 19, 2, 11, 20, 27, 2, 33, 2, 9, 30, 19, 16, 41, 2, 31, 28, 9, 2, 32, 16

Offset: 0

N. J. A. Sloane

nonn

The number of nonzero terms in the polynomial (1+x)^n (mod n). Note that n is prime iff a(n)=2. - T. D. Noe, Feb 23 2006

For n > 0: a(n) = number of terms > 0 in n-th row of triangle A053200. - Reinhard Zumkeller, Jan 24 2014

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A020475, A053200.

a007012 n = a007012_list !! n
a007012_list = 1 : map (sum . map signum) (tail a053200_tabl)
-- Reinhard Zumkeller, Jan 24 2014

Prepend[ Array[ Length[ Select[ Table[ Binomial[ #, k ]/#, {k, 0, #} ], !IntegerQ[ # ]& ] ]&, 100 ], 1 ]

a(n) = n + 1 - A020475(n). - T. D. Noe, Feb 23 2006

A094495 Table of binomial coefficients mod m^2, read by rows: T(m, n) = binomial(m, n) mod m^2.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 6, 56, 28, 8, 1, 1, 9, 36, 3, 45, 45, 3, 36, 9, 1, 1, 10, 45, 20, 10, 52, 10, 20, 45, 10, 1, 1, 11, 55, 44, 88, 99, 99, 88, 44, 55, 11, 1, 1, 12, 66, 76, 63, 72, 60, 72, 63, 76, 66, 12, 1

Offset: 0

Labos Elemer, Jun 02 2004

a(0) = 0 by convention.

			First deviation from A007318 is at a(40) = T(8,4) because binomial(8,4)=70 and 70 mod 64 = 6.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A007318, A053200, A094496, A094497.

Flatten[Table[Table[Mod[Binomial[n, j], n^2], {j, 0, n}], {n, 1, 20}], 1]

T(m,n)=binomial(m,n)%m^2 \\ Charles R Greathouse IV, Jul 29 2014

from math import comb, isqrt
def A094495(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2))%(r**2) if n else 0 # Chai Wah Wu, Apr 30 2025

A094496 Triangle read by rows: T(n,k) = binomial(n,k) - binomial(n,k) mod n^2, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 81, 81, 81, 81, 0, 0, 0, 0, 0, 0, 100, 200, 200, 200, 100, 0, 0, 0, 0, 0, 0, 121, 242, 363, 363, 242, 121, 0, 0, 0, 0, 0, 0, 144, 432, 720, 864, 720, 432, 144, 0, 0, 0

Offset: 0

Labos Elemer, Jun 02 2004

			Triangle begins:
  1;
  1, 1;
  0, 0, 0;
  0, 0, 0,   0;
  0, 0, 0,   0,   0;
  0, 0, 0,   0,   0,   0;
  0, 0, 0,   0,   0,   0,   0;
  0, 0, 0,   0,   0,   0,   0,   0;
  0, 0, 0,   0,  64,   0,   0,   0, 0;
  0, 0, 0,  81,  81,  81,  81,   0, 0, 0;
  0, 0, 0, 100, 200, 200, 200, 100, 0, 0, 0;
  ...
T(8,6) = binomial(8,4) - binomial(8,4) mod 8^2 = 70 - 6 = 64.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A007318, A053200, A094495, A094497.

Flatten[Table[Table[Binomial[n, j]-Mod[Binomial[n, j], n^2], {j, 0, n}], {n, 1, 20}], 1]

T(n,k) = my(x=binomial(n,k)); x - if(n, x % n^2) \\ Andrew Howroyd, Dec 12 2024

T(n,k) = A007318(n,k) - A094495(n,k).

cp's OEIS Frontend

A053204 Row sums of A053200.

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A020475 a(n) is the number of k for which binomial(n,k) is divisible by n.

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A053201 Pascal's triangle (excluding first, last element of each row) read by rows, row n read mod n.

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A053214 Central binomial coefficients (A000984) read mod 2n, with a(0)=1.

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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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A050870 T(h,k) = binomial(h,k) - A050186(h,k).

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