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 A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6. 8

%I

%S 1,0,1,0,1,1,1,1,1,1,2,2,2,2,3,3,4,4,4,5,5,6,7,7,8,9,10,11,13,13,15,

%T 16,17,20,21,23,25,27,30,33,36,38,42,45,49,54,57,62,67,72,79,85,92,98,

%U 106,114,123,133,141,152,163,175,189,202,216,231,248,265,284,304,323

%N Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

%C Also number of partitions of n into parts equal to 2,5, or 11 mod 12 (Gollnitz's theorem). Example: a(18)=4 because we have [14,2,2], [11,5,2], [5,5,2,2,2,2] and [2,2,2,2,2,2,2,2,2]. - _Emeric Deutsch_, Apr 18 2006

%H Alois P. Heinz, <a href="/A056970/b056970.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Alladi, <a href="http://at.yorku.ca/cgi-bin/amca/caew-39">Going beyond the partition theorem of Goellnitz</a>

%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/cbms/066">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 101.

%H G. E. Andrews, K. Alladi, B. Gordon, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002212471">Generalizations and refinements of a partition theorem of Göllnitz</a>, Journal für die reine und angewandte Mathematik (1995), Volume: 460, page 165-188.

%H H. Göllnitz, <a href="http://dx.doi.org/10.1515/crll.1967.225.154">Partitionen mit Differenzenbedingungen</a>, J. Reine Angew. Math. Vol. 225 (1967), 154-190.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoellnitzsTheorem.html"> Göllnitz's Theorem</a>

%F From _Emeric Deutsch_, Apr 18 2006: (Start)

%F G.f.: Product_{j >= 0} (1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)).

%F G.f.: 1/Product_{j >= 0} (1-x^(2+12j))(1-x^(5+12j))(1-x^(11+12j)).

%F (End)

%F Euler transform of period 12 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - _Michael Somos_, Jul 24 2007

%F a(n) ~ exp(Pi*sqrt(n/6)) / (2^(25/12) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015

%e a(18)=4 because we have [16,2], [14,4], [11,5,2] and [10,8].

%p g:=product((1+x^(2+6*j))*(1+x^(4+6*j))*(1+x^(5+6*j)),j=0..30): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..67); # _Emeric Deutsch_, Apr 18 2006

%p # second Maple program:

%p with(numtheory):

%p `if`(irem(d, 12) in [2, 5, 11], d, 0)

%p , d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Oct 27 2015

%t max = 70; g[x_] := Product[(1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), {j, 0, Floor[max/6]}]; CoefficientList[ Series[g[x], {x, 0, max}], x](* _Jean-François Alcover_, Nov 16 2011, after _Emeric Deutsch_ *)

%t a[n_] := a[n] = If[n==0, 1, Sum[Sum[If[MatchQ[Mod[d, 12], 2|5|11], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Dec 23 2015, after _Alois P. Heinz_ *)

%o (PARI) {a(n)= if(n<0, 0, polcoeff( 1/prod(k=1, n, 1-(k%3==2)*(k%12!=8)*x^k, 1+x*O(x^n)), n))} /* _Michael Somos_, Jul 24 2007 */

%o a056970 n = p a047261_list n where

%o p _ 0 = 1

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

%o -- _Reinhard Zumkeller_, Nov 16 2012

%Y Cf. A047261, A096981, A097451, A098884.

%K nonn,nice,easy

%O 0,11

%A _Eric W. Weisstein_

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Last modified August 20 03:33 EDT 2019. Contains 326139 sequences. (Running on oeis4.)