A053633 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1+x^j) mod x^(n+1)-1.

1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 3, 3, 3, 3, 6, 5, 5, 6, 5, 5, 10, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 30, 28, 28, 29, 28, 28, 29, 28, 28, 52, 51, 51, 51, 51, 52, 51, 51, 51, 51, 94, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 172, 170, 170, 172, 170, 170, 172

Offset

0,4

Comments

T(n,k) = number of binary vectors (x_1,...,x_n) satisfying Sum_{i=1..n} i*x_i = k (mod n+1) = size of Varshamov-Tenengolts code VT_k(n).

References

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.

Links

Seiichi Manyama, Rows n = 0..139, flattened

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].

N. J. A. Sloane, On single-deletion-correcting codes

Index entries for sequences related to subset sums modulo m

Index entries for sequences related to Gijswijt's sequence

Formula

The Maple code gives an explicit formula.

Example

Triangle begins:
  k  0    1    2    3    4    5    6    7    8    9
n
0    1;
1    1,   1;
2    2,   1,   1;
3    2,   2,   2,   2;
4    4,   3,   3,   3,   3;
5    6,   5,   5,   6,   5,   5;
6   10,   9,   9,   9,   9,   9,   9;
7   16,  16,  16,  16,  16,  16,  16,  16;
8   30,  28,  28,  29,  28,  28,  29,  28,  28;
9   52,  51,  51,  51,  51,  52,  51,  51,  51,  51;
    ...
[Edited by Seiichi Manyama, Mar 11 2018]

Maple

with(numtheory): A053633 := proc(n, k) local t1, d; t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+(1/(2*n))*2^(n/d)*phi(d)*mobius(d/gcd(d, k))/phi(d/gcd(d, k)); fi; od; t1; end;

Mathematica

Flatten[ Table[ CoefficientList[ PolynomialMod[ Product[1+x^j, {j, 1, n}], x^(n+1)-1], x], {n, 0, 11}]] (* Jean-François Alcover, May 04 2011 *)

Crossrefs

Cf. A053632, A063776, A300328, A300628. Leading coefficients give A000016, next column gives A000048.

Sequence in context: A134337 A261733 A268341 * A216460 A156755 A090822

Adjacent sequences: A053630 A053631 A053632 * A053634 A053635 A053636

Keyword

tabl,nonn,easy,nice

Author

N. J. A. Sloane, Mar 22 2000

Status

approved