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A210772 Number of partitions of 2^n into powers of 2 less than or equal to 8.

%I A210772 #21 Aug 27 2022 12:05:48
%S A210772 1,2,4,10,35,165,969,6545,47905,366145,2862209,22632705,180007425,
%T A210772 1435853825,11470030849,91693092865,733276217345,5865135816705,
%U A210772 46916791205889,375317149057025,3002468471537665,24019472891510785,192154683614691329,1537233070859485185
%N A210772 Number of partitions of 2^n into powers of 2 less than or equal to 8.
%H A210772 Alois P. Heinz, <a href="/A210772/b210772.txt">Table of n, a(n) for n = 0..1000</a>
%H A210772 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).
%F A210772 G.f.: -(12*x^5+11*x^4+30*x^3-44*x^2+13*x-1)/Product_{j=0..3} (2^j*x-1).
%F A210772 a(n) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..2} (1-x^(2^j)) for n>0.
%F A210772 a(n) = 1 + (11*2^(n-3))/3 + 2^(3*n-7)/3 + 4^(n-2) for n>1. - _Colin Barker_, Jan 26 2018
%e A210772 a(3) = 10 because there are 10 partitions of 2^3 = 8 into powers of 2 less than or equal to 8: [1,1,1,1,1,1,1,1], [2,1,1,1,1,1,1], [2,2,1,1,1,1], [2,2,2,1,1], [2,2,2,2], [4,1,1,1,1], [4,2,1,1], [4,2,2], [4,4], [8].
%p A210772 a:= n-> `if`(n<2, 2^n, (Matrix(4, (i, j)-> `if`(i=j-1, 1, `if`(i=4,
%p A210772      [-64, 120, -70, 15][j], 0)))^(n-2). <<4, 10, 35, 165>>)[1,1]):
%p A210772 seq(a(n), n=0..30);
%t A210772 LinearRecurrence[{15,-70,120,-64},{1,2,4,10,35,165},30] (* _Harvey P. Dale_, Aug 27 2022 *)
%o A210772 (PARI) Vec((1 - 13*x + 44*x^2 - 30*x^3 - 11*x^4 - 12*x^5) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 8*x)) + O(x^40)) \\ _Colin Barker_, Jan 26 2018
%Y A210772 Column k=3 of A152977.
%K A210772 nonn,easy
%O A210772 0,2
%A A210772 _Alois P. Heinz_, Mar 26 2012