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A133153 Number of partitions of n into parts that are odd or == +- 2 (mod 10). 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 21 15:46 EDT 2021. Contains 348155 sequences. (Running on oeis4.)