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Number of partitions of 2^n into powers of 2 less than or equal to 256.
2

%I #11 Jun 13 2015 00:54:13

%S 1,2,4,10,36,202,1828,27338,692004,30251721,2290267225,275723872209,

%T 45943934602273,9336623954364993,2119856439870545025,

%U 510453118614955153665,126696287737269468934657,31933986928271408429425665,8111646059635412792802330625

%N Number of partitions of 2^n into powers of 2 less than or equal to 256.

%H Alois P. Heinz, <a href="/A210777/b210777.txt">Table of n, a(n) for n = 0..130</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (511, -86870, 6304280, -211823808, 3389180928, -25822330880, 91089797120, -137170518016, 68719476736).

%F G.f.: (54717883351040*x^16 -83085001490432*x^15 +28916158300160*x^14 -281547988992*x^13 -272750006528*x^12 +100712240*x^11 -7148051274*x^10 -10790841321*x^9 -30886151190*x^8 +51093934102*x^7 -19831247382*x^6 +2989899926*x^5 -199557654*x^4 +6132574*x^3 -85852*x^2 +509*x-1) / Product_{j=0..8} (2^j*x-1).

%F a(n) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..7} (1-x^(2^j)) for n>0.

%p gf:= (-1 +(509 +(-85852 +(6132574 +(-199557654 +(2989899926 +(-19831247382 +(51093934102 +(-30886151190 +(-10790841321 +(-7148051274 +(100712240 +(-272750006528 +(-281547988992 +(28916158300160 +(-83085001490432 +54717883351040*x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x)/ mul(2^j*x-1, j=0..8): a:= n-> coeff(series(gf, x, n+1), x, n): seq(a(n), n=0..20);

%Y Column k=8 of A152977.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Mar 26 2012