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A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

%I A225045 #18 Jan 02 2017 17:45:48
%S A225045 1,0,1,0,1,1,1,2,1,3,1,4,3,5,5,5,7,7,10,10,13,13,16,18,21,25,27,32,33,
%T A225045 41,44,53,57,65,73,81,93,102,118,128,145,159,181,200,224,246,275,304,
%U A225045 337,375,413,460,503,559,614,679,749,821,907,991,1096,1197,1319,1442,1582,1733,1893,2076,2265,2482,2702,2956,3220
%N A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.
%H A225045 Alois P. Heinz and Vaclav Kotesovec, <a href="/A225045/b225045.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A225045 G.f.: prod(n>=1, 1 + q^n ) / prod(n>=1, 1 + q^(n*(n+1)/2) ). [_Joerg Arndt_, Apr 01 2014]
%F A225045 a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / (2+sqrt(2)) - 3*(3-2*sqrt(2)) * Zeta(3/2)^2 / (16*Pi)) / (2*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jan 02 2017
%e A225045 a(10) = #{8+2} = 1;
%e A225045 a(11) = #{11, 9+2, 7+4, 5+4+2} = 4;
%e A225045 a(12) = #{12, 8+4, 7+5} = 3;
%e A225045 a(13) = #{13, 11+2, 9+4, 8+5, 7+4+2} = 5.
%p A225045 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
%p A225045       `if`(n=0, 1, add(b(n-i*j, i-1), j=0..min(n/i,
%p A225045       `if`(issqr(8*i+1), 0, 1)))))
%p A225045     end:
%p A225045 a:= n-> b(n$2):
%p A225045 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 01 2014
%t A225045 b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n==0, 1, Sum[b[n-i*j, i-1], {j, 0, Min[n/i, If[IntegerQ[Sqrt[8*i+1]], 0, 1]]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jan 15 2016, after _Alois P. Heinz_ *)
%o A225045 (Haskell)
%o A225045 a225045 = p a014132_list where
%o A225045    p _      0 = 1
%o A225045    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%o A225045 (PARI) N=66; q='q+O('q^N); Vec( prod(n=1,N, 1 + q^n) / prod(n=1,N, 1 + q^(n*(n+1)/2)) ) \\ _Joerg Arndt_, Apr 01 2014
%Y A225045 Cf. A000009, A024940, A225044, A087154.
%K A225045 nonn
%O A225045 0,8
%A A225045 _Reinhard Zumkeller_, Apr 25 2013