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

0,5

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Case k=3,i=2 of Gordon/Goellnitz/Andrews Theorem.

Also number of partitions in which no odd part is repeated, with at most 1 part of size less than or equal to 2 and where differences between parts at distance 2 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.

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

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)

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A101195.

Sequence in context: A041003 A067592 A101195 * A123552 A071610 A358993

Adjacent sequences: A036015 A036016 A036017 * A036019 A036020 A036021

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved