A133153 Number of partitions of n into parts that are odd or == +- 2 (mod 10).

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 71, 90, 112, 140, 174, 214, 263, 322, 392, 476, 576, 694, 834, 1000, 1194, 1423, 1692, 2005, 2372, 2800, 3296, 3874, 4544, 5318, 6214, 7248, 8438, 9808, 11383, 13188, 15258, 17628, 20334, 23426, 26952, 30966, 35536, 40730

Offset

0,3

Comments

Andrews gives a second interpretation of these numbers and refers to them as a cousin of the Rogers-Ramanujan numbers.

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

Also number of partitions of n into parts not congruent to 0 or +-4 (mod 10). [Agarwal-Andrews (1987).] - N. J. A. Sloane, Nov 30 2019

References

Agarwal, A. K., and George E. Andrews. "Rogers-Ramanujan identities for partitions with 'N copies of N'." Journal of Combinatorial Theory, Series A 45.1 (1987): 40-49. See Theorem 1.

Links

Vincenzo Librandi, 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, Eq. (1.3). MR0858826 (88b:11063).

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 10.)

Mircea Merca, On the number of partitions into odd parts or congruent to +/-2 mod 10, Contributions to Discrete Mathematics, Vol. 13, No. 1 (2018).

Formula

Expansion of f(-q^4, -q^6) / f(-q) in powers of q where f() is Ramanujan's theta function.

Euler transform of period 10 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...]. - Michael Somos, Sep 30 2007

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

Example

G.f. = 1 + q + 2*q^2 + 3*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 11*q^7 + 15*q^8 + ...

Maple

lis1:=[1, 2, 3, 5, 7, 8, 9];
g1:= k -> 1/mul(1-x^(10*k+i), i in lis1);
g2:=mul(g1(k), k=0..100);
series(g2, x, 50); # N. J. A. Sloane, Nov 30 2019

Mathematica

th[m_][a_, b_] := Sum[a^(n*((n+1)/2))*b^(n*((n-1)/2)), {n, -m, m}]; th[m_][-q_] := th[m][-q, -q^2];
f[s_List] := (m++; CoefficientList[ Series[ th[m][-q^4, -q^6]/th[m][-q], {q, 0, 51}], q]); FixedPoint[f, m = 0; {}] (* Jean-François Alcover, Jul 19 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4, x^10] QPochhammer[ x^6, x^10] QPochhammer[ x^10] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Mar 27 2014 *)

Prog

(PARI) {a(n) = local(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) / (1 - x^k) / (1 - x^(2*k + 1)) + x * O(x^n), t), n))}; /* Michael Somos, Sep 30 2007 */

Crossrefs

Sequence in context: A175868 A035950 A175864 * A100673 A224216 A245432

Adjacent sequences: A133150 A133151 A133152 * A133154 A133155 A133156

Keyword

nonn

Author

N. J. A. Sloane, Sep 22 2007

Status

approved