A100853 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.
1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 31, 38, 48, 59, 72, 88, 107, 130, 157, 188, 225, 270, 321, 380, 451, 533, 627, 737, 864, 1011, 1181, 1375, 1599, 1858, 2152, 2488, 2875, 3316, 3816, 4387, 5036, 5773, 6610, 7555, 8626, 9840, 11207, 12748, 14489
Offset: 0
Crossrefs
Cf. A089958.
Programs
-
Maple
seq(coeff(mul((1+x^k)*(1+x^(4*k)),k=1..100),x,n),n=0..60); (C. Ronaldo)
-
Mathematica
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 *)
Formula
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)).
a(n) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 14 2018
Extensions
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
Comments