A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 21, 25, 30, 39, 46, 56, 72, 85, 101, 125, 147, 175, 215, 252, 296, 356, 415, 487, 582, 676, 786, 927, 1072, 1244, 1460, 1682, 1939, 2255, 2588, 2976, 3446, 3942, 4510, 5189, 5916, 6751, 7739, 8797, 9999, 11406, 12927, 14657

Offset

0,4

Comments

Also, number of partitions of in which the distinct parts are prime to 3 and the unrestricted parts are multiples of 3.

The inverted graded parafermionic partition function. This g.f. is a generalization of A003105, A006950 and A096938

References

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], (23-June-2013)

Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.

Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.

Formula

Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012

Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - Vladeta Jovovic, Aug 20 2004

G.f.: 1/product_{k>=1}(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)) = Product_{k>=1}(1+x^(3*k-1))(1+x^(3*k-2))/(1-x^(3*k)).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - Vaclav Kotesovec, Aug 31 2015

Example

a(11) = 15 because we can write 11 = 10+1 = 8+2+1 = 7+4 = 5+4+2 (parts do not contain multiple of 3) = 9+2 = 8+3 = 7+3+1 = 6+5 = 6+4+1 = 6+3+2 = 5+3+3 = 5+3+2+1 = 4+3+3+1 = 3+3+3+2.
1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + ...
q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

Maple

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)), k=1..150), x=0, 100);

Mathematica

CoefficientList[ Series[ Product[ 1/(1 - x^k + x^(2k) - x^(3k) + x^(4k) - x^(5k)), {k, 55}], {x, 0, 53}], x] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 100; CoefficientList[Series[x^3*QPochhammer[-1/x^2, x^3] * QPochhammer[-1/x, x^3]/((1 + x)*(1 + x^2) * QPochhammer[x^3, x^3]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / (eta(x + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */
(Haskell)
a096981 = p $ tail a047273_list where
   p _  0         = 1
   p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

Crossrefs

Cf. A047273, A056970, A097451, A098884.

Sequence in context: A006065 A218933 A266746 * A281966 A276431 A308272

Adjacent sequences: A096978 A096979 A096980 * A096982 A096983 A096984

Keyword

nonn

Author

Noureddine Chair, Aug 19 2004

Extensions

Better definition from Vladeta Jovovic, Aug 20 2004

More terms from Robert G. Wilson v, Aug 21 2004

Incorrect b-file replaced by Vaclav Kotesovec, Aug 31 2015

Status

approved