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%I #13 Aug 14 2018 10:50:31
%S 1,1,1,2,3,4,5,7,9,12,15,19,25,31,38,48,59,72,88,107,130,157,188,225,
%T 270,321,380,451,533,627,737,864,1011,1181,1375,1599,1858,2152,2488,
%U 2875,3316,3816,4387,5036,5773,6610,7555,8626,9840,11207,12748,14489
%N Number of partitions of n in which every part occurs 1, 4, or 5 times. Also number of partitions of n in which every part is congruent to {1, 3, 4, 5, 7} mod 8.
%C Also number of partitions of n in which every even part occurs exactly twice. - _Vladeta Jovovic_, Oct 06 2007
%F Euler transform of period 8 sequence [1, 0, 1, 1, 1, 0, 1, 0, ...]. G.f.: Product_{k>0} (1+x^k)*(1+x^(4*k)) = 1/Product_{k>0} (1-x^A047501(k)).
%F a(n) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (8 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 14 2018
%p seq(coeff(mul((1+x^k)*(1+x^(4*k)),k=1..100),x,n),n=0..60); (C. Ronaldo)
%t np145Q[j_]:=SubsetQ[{1,4,5},Union[Tally[j][[All,2]]]]; Table[Length[ Select[ IntegerPartitions[n],np145Q]],{n,0,51}] (* _Harvey P. Dale_, Aug 04 2018 *)
%Y Cf. A089958.
%K easy,nonn
%O 0,4
%A _Vladeta Jovovic_, Jan 08 2005
%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
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