0,2

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).

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

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

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).