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

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

Author

Labos Elemer, Jun 02 2004

Keywords

• easy
• nonn
• tabl

Comments

a(0) = 0 by convention.

Examples

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

Links

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

Crossrefs

Cf. A007318, A053200, A094496, A094497.

Programs

Mathematica

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

PARI

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

Python

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