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 A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6. 9

%I

%S 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,

%T 107,142,143,188,191,247,253,321,332,418,432,537,561,690,721,880,924,

%U 1118,1178,1412,1493,1781,1884,2231,2370,2789,2965,3472,3698,4309,4596

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

%C 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

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

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

%H M. Somos, <a href="http://somos.crg4.com/multiq.html">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

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

%F Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3) in powers of q. - _Michael Somos_, Sep 24 2013

%F a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015

%F 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

%e a(8)=5 because we have [8],[44],[422],[332] and [2222].

%e G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...

%e G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...

%p 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

%t a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}; (* _Michael Somos_, Sep 24 2013 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Sep 24 2013 *)

%o a097451 n = p a047228_list n where

%o p _ 0 = 1

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

%o -- _Reinhard Zumkeller_, Nov 16 2012

%o (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 */

%o (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 */

%Y Cf. A070047.

%Y Cf. A047228, A056970, A096981, A098884.

%K easy,nonn

%O 0,5

%A _Vladeta Jovovic_, Aug 23 2004

%E More terms from _Emeric Deutsch_, Feb 16 2006

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Last modified September 20 03:55 EDT 2019. Contains 327212 sequences. (Running on oeis4.)