A094497 -id:A094497

Search: a094497 -id:a094497

Displaying 1-2 of 2 results found.

page 1

A094495

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

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

a(0) = 0 by convention.

LINKS

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A007318, A053200, A094496, A094497.

KEYWORD

easy,nonn,tabl

AUTHOR

Labos Elemer, Jun 02 2004

STATUS

approved

A094496

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

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,41

LINKS

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

FORMULA

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

EXAMPLE

Triangle begins:

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.

MATHEMATICA

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

PROG

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