0,2

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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]

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [ Swedish ], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.

R. F. Churchhouse, Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66 1969 371-376.

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.

C.-E. Froberg, Accurate estimation of the number of binary partitions, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), 386-391.

H. Minc, The free commutative entropic logarithmetic. Proc. Roy. Soc. Edinburgh Sect. A 65 1959 177-192 (1959).

V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp.17-41 [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2009]

R. F. Churchhouse, Congruence properties of the binary partition function, Math. Proc. Cambr. Phil. Soc. vol 66, no. 2 (1969), 365-370. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2009]

Carl-Erik Froberg, Accurate estimation of the number of binary partitions, BIT Numerical Mathematics vol. 17, no 4 (1977) 386-391. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2009]

V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

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

a(n) = A078121(n+1, 1). - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 13 2004

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]

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]

A002577 := proc(n) if n<=1 then n+1 else A000123(2^(n-1)); fi; end;

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]

(PARI) a(n)=polcoeff(prod(j=0, n, 1/(1-x^(2^j)+x*O(x^(2^n)))), 2^n) (Paul D. Hanna)

a(n)=A000123(2^(n-1))=A018818(2^n).

Cf. A078537.

Cf. A078121.

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]

Sequence in context: A038077 A006396 A066278 this_sequence A076132 A047142 A081080

Adjacent sequences: A002574 A002575 A002576 this_sequence A002578 A002579 A002580

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).