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A179046 Partitions into distinct parts with minimal difference 3 and minimal part >= 3.

%I A179046 #34 Nov 20 2020 05:53:47
%S A179046 1,0,0,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,7,8,9,11,12,14,16,18,20,23,25,28,
%T A179046 32,35,39,44,49,54,61,67,75,83,92,101,113,123,136,150,165,180,199,217,
%U A179046 239,261,286,312,343,373,408,445,486,528,577,626,682,741,805,873,949,1027,1114
%N A179046 Partitions into distinct parts with minimal difference 3 and minimal part >= 3.
%H A179046 Alois P. Heinz, <a href="/A179046/b179046.txt">Table of n, a(n) for n = 0..1000</a>
%F A179046 G.f.: sum(n>=0, x^(3*n*(n+1)/2) / prod(k=1,n,1-x^k) ), this is a special case of the g.f. sum(n>=0, x^(D*n*(n+1)/2) / prod(k=1,n,1-x^k) ) for partitions into distinct parts with minimal difference D and minimal part >= D. - _Joerg Arndt_, Apr 07 2011
%F A179046 The g.f. for partitions into distinct parts with minimal difference D and no restriction on the minimal part is sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ). - _Joerg Arndt_, Mar 31 2014
%e A179046 a(13)=4 because there are 4 such partitions of 13: 3+10=4+9=5+8=13.
%e A179046 a(0)=1 because the condition is void for the empty list.
%p A179046 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A179046       `if`(n>i*(i+1)/2-3, 0, b(n, i-1)+
%p A179046       `if`(i>n, 0, b(n-i, i-3))))
%p A179046     end:
%p A179046 a:= n-> b(n$2):
%p A179046 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A179046 b[n_, i_] := b[n, i] = If[n == 0, 1,
%t A179046   If[n > i(i+1)/2 - 3, 0, b[n, i - 1] +
%t A179046   If[i > n, 0, b[n - i, i - 3]]]];
%t A179046 a[n_] := b[n, n];
%t A179046 a /@ Range[0, 80] (* _Jean-François Alcover_, Nov 20 2020, after _Alois P. Heinz_ *)
%o A179046 (Sage)
%o A179046 A179046 = lambda n: Partitions(n,max_slope=-3).filter(lambda x: not x or min(x) >= 3).cardinality() # _D. S. McNeil_, Jan 04 2011
%o A179046 (PARI)
%o A179046 N=66; x='x+O('x^N);
%o A179046 gf = sum(n=0,N, x^(3*n*(n+1)/2)/prod(k=1,n,1-x^k));
%o A179046 v = Vec(gf)
%o A179046 /* _Joerg Arndt_, Apr 07 2011 */
%Y A179046 Cf. A003106 (min diff=2, min part=2), A000009 (min diff=1, min part=1).
%Y A179046 Cf. A003114 (min diff=2), A025157 (min diff=3), A025158 (min diff=4), A025159 (min diff=5), A025160 (min diff=6), A025161 (min diff=7), A025162 (min diff=8).
%K A179046 nonn
%O A179046 0,10
%A A179046 _Joerg Arndt_, Jan 04 2011