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Number of partitions of 2^n into powers of 2 less than or equal to 1024.
2

%I #9 Jun 13 2015 00:54:13

%S 1,2,4,10,36,202,1828,27338,692004,30251722,2320518948,316359580361,

%T 77160820913241,31769732129318865,19210889607930498081,

%U 14781930262928342616641,13037860166110209522457729,12369535268518332988593592577,12186672180675798897571822711297

%N Number of partitions of 2^n into powers of 2 less than or equal to 1024.

%H Alois P. Heinz, <a href="/A210779/b210779.txt">Table of n, a(n) for n = 0..110</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2047, -1396054, 407647768, -55440096448, 3634008902656, -116288284884992, 1816661080408064, -13678389311307776, 47968050187599872, -72022409665839104, 36028797018963968).

%F G.f.: (-95460882767931143880704*x^20 +143817459476148640546816*x^19 -47353367247905900986368*x^18 -1833431416452222550016*x^17 +851800662334219223040*x^16 -21573396097321885696*x^15 -5556995021730048*x^14 -2458328204903632*x^13 -3738333173327178*x^12 -5646507818862569*x^11 -16176670881134614*x^10 +26835366859855894*x^9 -10475796196345878*x^8 +1598080542315542*x^7 -109238000834070*x^6 +3524745413782*x^5 -54630364694*x^4 +404863838*x^3 -1391964*x^2 +2045*x-1) / Product_{j=0..10} (2^j*x-1).

%F a(n) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..9} (1-x^(2^j)) for n>0.

%p gf:= (-1 +(2045 +(-1391964 +(404863838 +(-54630364694 +(3524745413782 +(-109238000834070 +(1598080542315542 +(-10475796196345878 +(26835366859855894 +(-16176670881134614 +(-5646507818862569 +(-3738333173327178 +(-2458328204903632 +(-5556995021730048 +(-21573396097321885696 +(851800662334219223040 +(-1833431416452222550016 +(-47353367247905900986368

%p +(143817459476148640546816 -95460882767931143880704*x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x) *x)/ mul(2^j*x-1, j=0..10): a:= n-> coeff(series(gf, x, n+1), x, n): seq(a(n), n=0..20);

%Y Column k=10 of A152977.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Mar 26 2012