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A007730 7th binary partition function.

%I A007730 #30 Feb 20 2021 03:35:58
%S A007730 1,1,2,2,4,4,6,5,9,8,12,10,16,14,19,15,24,20,28,22,34,29,39,30,46,38,
%T A007730 52,40,59,49,64,48,72,58,78,59,87,72,94,70,104,84,113,85,124,102,132,
%U A007730 98,144,115,153,114,166,136,176,130,189,151,200,148,212,172,220
%N A007730 7th binary partition function.
%H A007730 Alois P. Heinz, <a href="/A007730/b007730.txt">Table of n, a(n) for n = 0..10000</a>
%H A007730 B. Reznick, <a href="http://dx.doi.org/10.1007/978-1-4612-3464-7_29">Some binary partition functions</a>, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
%F A007730 G.f.: Product_{k>=0} (1 - x^(7*2^k))/(1 - x^(2^k)). - _Ilya Gutkovskiy_, Jul 09 2019
%p A007730 b:= proc(n, i) option remember;
%p A007730       `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
%p A007730          b(n-j*2^i, i-1)), j=0..6)))
%p A007730     end:
%p A007730 a:= n-> b(n, ilog2(n)):
%p A007730 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jun 21 2012
%t A007730 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0,
%t A007730      Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]];
%t A007730 a[n_] := b[n, Length[IntegerDigits[n, 2]] - 1, 7];
%t A007730 Table[a[n], {n, 0, 70} ] (* _Jean-François Alcover_, Jan 17 2014, after _Alois P. Heinz_ *)
%Y A007730 A column of A072170.
%K A007730 nonn
%O A007730 0,3
%A A007730 _N. J. A. Sloane_
%E A007730 More terms from _Vladeta Jovovic_, May 07 2004