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A276431 Number of partitions of n containing no parts that are powers of 2 with positive exponent. 8

%I

%S 1,1,1,2,2,3,5,6,7,10,13,16,22,27,33,43,52,64,81,98,120,148,178,215,

%T 263,315,377,455,541,644,771,912,1078,1279,1506,1772,2089,2447,2864,

%U 3356,3916,4563,5320,6180,7171,8324,9633,11136,12874,14845,17102,19696

%N Number of partitions of n containing no parts that are powers of 2 with positive exponent.

%H Alois P. Heinz, <a href="/A276431/b276431.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A276430(n,0).

%F G.f.: g(x) = Product_{i>=1} (1-x^{h(i)})/(1-x^i), where h(i) = 2^i.

%e a(6) = 5, counting [1,1,1,1,1,1], [1,1,1,3], [1,5], [3,3], [6].

%p h:= proc(i) options operator, arrow: 2^i end proc: g := product((1-x^h(i))/(1-x^i), i = 1 .. 55): gser := series(g, x = 0, 55): seq(coeff(gser, x, n), n = 0 .. 50);

%p # second Maple program:

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d>1 and

%p d=2^ilog2(d), 0, a(n-j)*d), d=divisors(j)), j=1..n)/n)

%p end:

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Sep 20 2016

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d>1 && d == 2^Floor[Log[2, d]], 0, a[n-j]*d], {d, Divisors[j]}], {j, 1, n}]/n]; Table[a[n], {n, 0, 55}] (* _Jean-Fran├žois Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)

%Y Cf. A276430.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Sep 19 2016

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Last modified March 31 20:59 EDT 2023. Contains 361673 sequences. (Running on oeis4.)