A096981 - OEIS

%I #40 Nov 28 2024 13:27:05

%S 1,1,1,2,2,3,5,6,7,10,12,15,21,25,30,39,46,56,72,85,101,125,147,175,

%T 215,252,296,356,415,487,582,676,786,927,1072,1244,1460,1682,1939,

%U 2255,2588,2976,3446,3942,4510,5189,5916,6751,7739,8797,9999,11406,12927,14657

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

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

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

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

%H Vaclav Kotesovec, <a href="/A096981/b096981.txt">Table of n, a(n) for n = 0..10000</a>

%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.

%H Noureddine Chair, <a href="http://arxiv.org/abs/1306.5415">The Euler-Riemann Gases, and Partition Identities</a>, arXiv:1306.5415 [math-ph], 23-June-2013.

%H Donald Spector, <a href="http://arxiv.org/abs/hep-th/9710002">Duality, partial supersymmetry and arithmetic number theory</a>, arXiv:hep-th/9710002, 1997.

%H Donald Spector, <a href="http://dx.doi.org/10.1063/1.532269">Duality, partial supersymmetry and arithmetic number theory</a>, J. Math. Phys. Vol. 39, 1998, p. 1919.

%F Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - _Michael Somos_, Jun 08 2012

%F Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - _Vladeta Jovovic_, Aug 20 2004

%F 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)).

%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - _Vaclav Kotesovec_, Aug 31 2015

%e 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.

%e 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 + ...

%e q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

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

%t 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 *)

%t 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 *)

%o (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 */

%o (Haskell)

%o a096981 = p $ tail a047273_list where

%o    p _  0         = 1

%o    p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Feb 19 2013

%Y Cf. A047273, A056970, A097451, A098884.

%K nonn

%O 0,4

%A _Noureddine Chair_, Aug 19 2004

%E Better definition from _Vladeta Jovovic_, Aug 20 2004

%E More terms from _Robert G. Wilson v_, Aug 21 2004

%E Incorrect b-file replaced by _Vaclav Kotesovec_, Aug 31 2015