A036018 Number of partitions of n into parts not of form 4k+2, 12k, 12k+3 or 12k-3.
1, 1, 1, 1, 2, 3, 3, 4, 6, 7, 8, 10, 13, 16, 18, 22, 28, 33, 38, 45, 55, 65, 74, 87, 104, 121, 138, 160, 188, 217, 247, 284, 330, 378, 428, 489, 562, 640, 722, 820, 936, 1059, 1191, 1345, 1524, 1717, 1924, 2163, 2438, 2734, 3054, 3419, 3834, 4284, 4770, 5321, 5943
Offset: 0
Examples
1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ... q + q^5 + q^9 + q^13 + 2*q^17 + 3*q^21 + 3*q^25 + 4*q^29 + 6*q^33 + ...
References
- G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
- G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
- Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S27.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Cf. A101195.
Programs
-
Mathematica
nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k - 5))*(1 + x^(6*k - 1))/((1 - x^(6*k - 4))*(1 - x^(6*k - 2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2017 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ([1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0][(k-1)%12 + 1]) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jun 28 2004 */ -
PARI
{a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 28 2004 */
Formula
Expansion of q^(-1/4) * (eta(q^2) * eta(q^3) * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Jun 28 2004
Euler transform of period 12 sequence [1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, ...]. - Michael Somos, Jun 28 2004
Expansion of psi(-x^3) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 21 2007
Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = u^3 * (1 + v^4) - v * (1 + u*v)^3. - Michael Somos, Nov 21 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = sqrt(1 / 3) / f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 21 2007
A101195(n) = (-1)^n * a(n).
From Vaclav Kotesovec, Jan 12 2017: (Start)
a(n) ~ Pi * BesselI(1, sqrt(4*n+1)*Pi/(2*sqrt(3))) / (3*sqrt(4*n+1)).
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)) * (1 + (Pi/24 - 3/(8*Pi))*sqrt(3/n) + (Pi^2/384 - 45/(128*Pi^2) - 15/64)/n).
(End)
Comments