Andrew Howroyd, <a href="/A094496/b094496_1.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

proposed

approved

editing

proposed

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

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;

...

a[n]=C[n,j]-Mod[C[n,j],n^2], difference of binomial coefficients and its residue mod n^2 read by rows.

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

a[0]=1 by convention.

First deviation from A007318 is at a[40] because C[8,4]=70 and Mod[70,64]=6.

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

T(8,6) = binomial(8,4) - binomial(8,4) mod 8^2 = 70 - 6 = 64.

First deviation from A007318 is at a[40] because C[8,4]=70 and Mod[70,64]=6.

Andrew Howroyd, <a href="/A094496/b094496_1.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

First deviation from A007318 is at a[40] because C[8,4]=70 and Mod[70,64]=6.

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

easy,nonn,tabl,less

approved

editing

_Labos E. (labos(AT)ana.sote.hu), Elemer_, Jun 02 2004

First deviation from A007318 is at a[40], because C[8,4]=70 and Mod[70,64]=6.

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

easy,nonn,tabl,new

easy,nonn,tabl,new

Labos E. (labos(AT)ana1ana.sote.hu), Jun 02 2004

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

1,40

0,41

a[0]=1 by convention.

easy,nonn,tabl,new