A101417 - OEIS

%I #13 Jan 16 2019 20:30:55

%S 1,0,0,1,0,1,2,1,1,3,3,3,6,5,6,10,9,12,17,17,22,28,30,37,48,52,62,78,

%T 86,103,127,141,166,201,227,266,317,358,417,492,560,647,757,860,991,

%U 1153,1309,1503,1738,1971,2257,2594,2941,3356,3843,4351,4948,5644,6382,7240

%N Number of partitions of n into parts without powers of 2.

%H Alois P. Heinz, <a href="/A101417/b101417.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Product_{j>=1} (1-x^(2^j)) / Product_{i>=2} (1-x^i). - _Emeric Deutsch_, Mar 29 2006

%e a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.

%e From _Gus Wiseman_, Jan 07 2019: (Start)

%e The a(3) = 1 through a(14) = 5 integer partitions (A = 10, ..., E = 14):

%e   (3)  (5)  (6)   (7)  (53)  (9)    (A)   (B)    (C)     (D)    (E)

%e             (33)             (63)   (55)  (65)   (66)    (76)   (77)

%e                              (333)  (73)  (533)  (75)    (A3)   (95)

%e                                                  (93)    (553)  (B3)

%e                                                  (633)   (733)  (653)

%e                                                  (3333)         (5333)

%e (End)

%p g:= product(1-x^(2^j),j=0..15)/product(1-x^i,i=1..75): gser:= series(g, x=0,62): seq(coeff(gser,x,n),n=0..59); # _Emeric Deutsch_, Mar 29 2006

%t Table[Length[Select[IntegerPartitions[n],And@@Not/@IntegerQ/@Log[2,#]&]],{n,20}] (* _Gus Wiseman_, Jan 07 2019 *)

%Y Cf. A000041, A018819, A000123.

%Y Cf. A087897, A102430, A276431, A321346, A323053, A323092, A323093.

%K nonn

%O 0,7

%A _Reinhard Zumkeller_, Jan 16 2005