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
Examples
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]
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
Crossrefs
Programs
-
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 *)
Formula
The Maple code gives an explicit formula.
Comments