A134157 Number of partitions of n into parts that are odd or == +- 4 mod 10.

1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 30, 38, 48, 61, 76, 94, 117, 144, 176, 216, 262, 317, 384, 462, 554, 664, 792, 942, 1120, 1326, 1566, 1848, 2174, 2552, 2992, 3499, 4084, 4762, 5540, 6434, 7464, 8642, 9991, 11538, 13302, 15314, 17612, 20225, 23196

Offset

0,4

Comments

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 7, Equ. (1.4). MR0858826 (88b:11063).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x^2, -x^8) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

Euler transform of period 10 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].

G.f.: ( Sum_{k>=0} x^(2*(k^2+k)) / ((1 - x^2) * (1 - x^4) * ... * (1 - x^(2*k))) ) / Product_{k>0} 1 - x^(2*k-1).

G.f.: Sum_{k>=0} x^(k*(3*k+3)/2) * (1 + x) * (1 + x^2) * ... * (1 + x^(2*k)) / ( (1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1)) ).

Example

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
G.f. = q^49 + q^169 + q^289 + 2*q^409 + 3*q^529 + 4*q^649 + 6*q^769 + 8*q^889 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10] QPochhammer[ x^10] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Oct 26 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Oct 26 2015 *)

Prog

(PARI) {a(n) = my(t); if( n<0, 0, t = 1 / (1 - x) + x * O(x^n); polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1)\6, t = t * x^(3*k) / ((1 - x^k) * (1 - x^(2*k + 1))) + x * O(x^n), t), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1] * x^k, 1 + x * O(x^n)), n))};

Crossrefs

Sequence in context: A095913 A102848 A260183 * A274194 A237751 A045476

Adjacent sequences: A134154 A134155 A134156 * A134158 A134159 A134160

Keyword

nonn

Author

Michael Somos, Oct 10 2007

Status

approved