A094495 -id:A094495 - cp's OEIS frontend

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

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).

A094497 Triangular table A(n,j) = C(n,j) - C(n,j) mod n^3, difference of binomial coefficient and its residue mod n^3, read by rows.

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, 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, 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, 2744, 2744, 2744, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3375, 3375, 3375, 3375, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096, 4096, 8192, 12288, 8192, 4096, 4096

Offset: 0

Labos Elemer, Jun 02 2004

a(0) = 1 by convention.

			Out of a(1)=1 and a(2)=1, the first deviation from A007318 is at a(111) because C(14,6) = 3003, 3003 mod 2744 = 259, a(111) = 2744.

Cf. A007318, A053200, A094495, A094496.

Flatten[Table[Table[Binomial[n, j]-Mod[Binomial[n, j], n^3], {j, 0, n}], {n, 1, 14}], 1]

cp's OEIS Frontend

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

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