|
Search: id:A122134
|
|
|
| A122134 |
|
Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))). |
|
+0
2
|
|
| 1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 18, 24, 28, 36, 42, 54, 62, 78, 91, 112, 130, 159, 184, 222, 258, 308, 356, 424, 488, 576, 664, 778, 894, 1044, 1196, 1389, 1590, 1838, 2098, 2419, 2754, 3162, 3596, 4114, 4668, 5328, 6032, 6864, 7760, 8806, 9936, 11252
(list; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
|
|
REFERENCES
|
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)
|
|
FORMULA
|
Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
G.f.: Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).
Expansion of f(-q,-q^9) * f(-q^8,-q^12) / ( f(-q) * f(-q^20) ) in powers of q where f(-q) := f(-q,-q^2) and f(a,b) is Ramanujan's two variable theta function.
|
|
PROGRAM
|
(PARI) {a(n)=if(n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)/ prod(i=1, 2*k, 1-x^i, 1+x*O(x^(n-k^2-k)))), n))}
|
|
CROSSREFS
|
Sequence in context: A029008 A136343 A001996 this_sequence A035940 A067772 A058686
Adjacent sequences: A122131 A122132 A122133 this_sequence A122135 A122136 A122137
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Michael Somos, Aug 21 2006, Oct 10 2007
|
|
|
Search completed in 0.002 seconds
|
