A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

1, 0, 1, 1, 2, 1, 3, 2, 5, 4, 7, 6, 11, 9, 15, 14, 22, 20, 31, 29, 43, 41, 58, 57, 80, 78, 106, 107, 142, 143, 188, 191, 247, 253, 321, 332, 418, 432, 537, 561, 690, 721, 880, 924, 1118, 1178, 1412, 1493, 1781, 1884, 2231, 2370, 2789, 2965, 3472, 3698, 4309, 4596

Offset

0,5

Comments

Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - Emeric Deutsch, Feb 16 2006

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

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.

Links

Table of n, a(n) for n=0..57.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].

G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - Emeric Deutsch, Feb 16 2006

Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 24 2013

Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3) in powers of q. - Michael Somos, Sep 24 2013

a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015

Example

a(8)=5 because we have [8],[44],[422],[332] and [2222].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...

Maple

g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)), j=0..15): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=0..67); # Emeric Deutsch, Feb 16 2006

Mathematica

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}; (* Michael Somos, Sep 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)

Prog

(Haskell)
a097451 n = p a047228_list n where
   p _  0         = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 24 2013 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 24 2013 */

Crossrefs

Cf. A070047.

Cf. A047228, A056970, A096981, A098884.

Sequence in context: A238782 A378749 A058736 * A005916 A034392 A361392

Adjacent sequences: A097448 A097449 A097450 * A097452 A097453 A097454

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 23 2004

Extensions

More terms from Emeric Deutsch, Feb 16 2006

Status

approved