%I A002577 M1239 N0473
%S A002577 1,2,4,10,36,202,1828,27338,692004,30251722,2320518948,316359580362,
%T A002577 77477180493604,34394869942983370,27893897106768940836,41603705003444309596874,
%U A002577 114788185359199234852802340,588880400923055731115178072778
%N A002577 Number of partitions of 2^n into powers of 2.
%D A002577 Bakoev V., Algorithmic approach to counting of certain types m-ary partitions, 
               Discrete Mathematics, 275 (2004) pp.17-41. [From Valentin Bakoev 
               (v_bakoev(AT)yahoo.com), Feb 25 2009]
%D A002577 G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [ Swedish 
               ], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.
%D A002577 R. F. Churchhouse, Congruence properties of the binary partition function. 
               Proc. Cambridge Philos. Soc. 66 1969 371-376.
%D A002577 R. F. Churchhouse, Binary partitions, pp. 397-400 of A. O. L. Atkin and 
               B. J. Birch, editors, Computers in Number Theory. Academic Press, 
               NY, 1971.
%D A002577 C.-E. Froberg, Accurate estimation of the number of binary partitions, 
               Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), 386-391.
%D A002577 H. Minc, The free commutative entropic logarithmetic. Proc. Roy. Soc. 
               Edinburgh Sect. A 65 1959 177-192 (1959).
%D A002577 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002577 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002577 V. Bakoev, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00096-7">
               Algorithmic approach to counting of certain types m-ary partitions</
               a>, Discrete Mathematics, 275 (2004) pp.17-41 [From R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Apr 20 2009]
%H A002577 R. F. Churchhouse, <a href="http://dx.doi.org/10.1017/S0305004100045072">
               Congruence properties of the binary partition function</a>, Math. 
               Proc. Cambr. Phil. Soc. vol 66, no. 2 (1969), 365-370. [From R. J. 
               Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2009]
%H A002577 Carl-Erik Froberg, <a href="http://dx.doi.org/10.1007/BF01933448">Accurate 
               estimation of the number of binary partitions</a>, BIT Numerical 
               Mathematics vol. 17, no 4 (1977) 386-391. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 20 2009]
%H A002577 V. Bakoev, <a href="http://www.uni-vt.bg/showfullpub.asp?u=32&fn=AACCTMP.pdf">
               Algorithmic approach to counting of certain types m-ary partitions</
               a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%F A002577 a(n) is about 0.9233*sum_j {i=0, 1, 2, 3, ...} 2^(j*(2n-j-1)/2)/j!. - 
               Henry Bottomley (se16(AT)btinternet.com), Jul 23 2003
%F A002577 a(n) = A078121(n+1, 1). - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 
               13 2004
%F A002577 Denote the sum: m^n+m^n+...+m^n, k times, by k.m^n (m>1, n>0 and k are 
               natural numbers). The general formula for the number of all partitions 
               of the sum k.m^n into powers of m is: t_m(n, k)= k+1 if n=1, t_m(n, 
               k)= 1 if k=0, and t_m(n, k)= t_m(n, k-1) + t_m(n-1, k.m) if n>1 and 
               k>0. A002577 is obtained for m=2 and n=1,2,3,... [From Valentin Bakoev 
               (v_bakoev(AT)yahoo.com), Feb 25 2009]
%e A002577 To compute t_2(6,1) we can use a table T, defined as T[i,j]= t_2(i,j), 
               for i=1,2,...,6(=n), and j= 0,1,2,...,32(= k.m^{n-1}). It is: 1,2,
               3,4,5,6,7,8,9...,33; 1,4,9,16,25,36,49...,81; (so the second row 
               contains the first members of A000290 - the square numbers) 1,10,
               35,84,165,...,969; (so the third row contains the first members of 
               A000447. The r-th tetrahedral number is given by formula r(r+1)(r+2)/
               6. This row (also A000447) contains the tetrahedral numbers, obtained 
               for r=1,3,5,7,...) 1,36,201,656,1625; 1,202,1827; 1,1828; Column 
               1 contains the first 6 members of A002577. [From Valentin Bakoev 
               (v_bakoev(AT)yahoo.com), Feb 25 2009]
%p A002577 A002577 := proc(n) if n<=1 then n+1 else A000123(2^(n-1)); fi; end;
%p A002577 There exists an algorithm (with polynomial running-time) for computing 
               the members of A002577, A125792 and other sequences of the same type. 
               [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 25 2009]
%o A002577 (PARI) a(n)=polcoeff(prod(j=0,n,1/(1-x^(2^j)+x*O(x^(2^n)))),2^n) (Paul 
               D. Hanna)
%Y A002577 a(n)=A000123(2^(n-1))=A018818(2^n).
%Y A002577 Cf. A078537.
%Y A002577 Cf. A078121.
%Y A002577 1) Subtracting 1 from the members of A002577 we obtain A125892; 2) A002577 
               is related to A000290 and to A000447, as it is shown in the example; 
               3) For given m, the general formula for t_m(n, k) and the corresponding 
               tables T, computed as in the example, determine a family of related 
               sequences (placed in the rows or in the columns of T). For example, 
               the numbers from the second row of T, computed for given m and n>
               2, are the (m+2)-gonal numbers. So the second row contains the first 
               members of: A000290 (the square numbers) when m=2, A000326 (the pentagonal 
               numbers) when m=3, and so on. But the numbers from IV, V, and etc. 
               rows of the given table are not represented in OEIS till now. [From 
               Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 25 2009]
%Y A002577 Sequence in context: A038077 A006396 A066278 this_sequence A076132 A047142 
               A081080
%Y A002577 Adjacent sequences: A002574 A002575 A002576 this_sequence A002578 A002579 
               A002580
%K A002577 nonn,easy,nice
%O A002577 0,2
%A A002577 N. J. A. Sloane (njas(AT)research.att.com).