0,3

Equals the number of partitions of 2^n-1 into powers of 2 (cf. A018819). a(n) = A018819(2^n-1) = binary partitions of 2^n-1. - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 22 2004

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).

George E. Andrews, Peter Paule, Axel Riese and Volker Strehl, "MacMahon's Partition Analysis V: Bijections, recursions and magic squares," in Algebraic Combinatorics and Applications, edited by Anton Betten, Axel Kohnert, Reinhard Laue and Alfred Wassermann [Proceedings of ALCOMA, September 1999] (Springer, 2001), 1-39.

A. Cayley, "On a problem in the partition of numbers," Philosophical Magazine (4) 13 (1857), 245-248; reprinted in his Collected Math. Papers, Vol. 3, pp. 247-249

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.

D. E. Knuth, Selected Papers on Analysis of Algorithms, p. 75 (gives asymptotic formula and lower bound).

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

T. K. Moon (tmoon(AT)artemis.ece.usu.edu), Enumerations of binary trees, types of trees and the number of reversiblevariable length codes, submitted to Discrete Applied Mathematics, 2000.

T. D. Noe, Table of n, a(n) for n=0..50

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

a(n) = A098539(n, 1). - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 13 2004

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1) + xF(x,2n) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007

d := proc(n) option remember; if n=0 then 1 else sum(d(n-1), k=1..2*k) fi end; A002449 := n -> eval(d(n-1), k=1);

(PARI) {a(n)=local(A, B, C, m); A=matrix(1, 1); A[1, 1]=1; for(m=2, n+1, B=A^2; C=matrix(m, m); for(j=1, m, for(k=1, j, if(j<3|k==j|k>m-1, C[j, k]=1, if(k==1, C[j, k]=B[j-1, 1], C[j, k]=B[j-1, k-1])); )); A=C); A[n+1, 1]} (Paul Hanna)

(PARI) a(n)=polcoeff(1/prod(k=0, n, 1-x^(2^k)+O(x^(2^n))), 2^n-1)

Cf. A001699, A056207.

Cf. A098539.

Cf. A018819.

Sequence in context: A047863 A141713 A005272 this_sequence A059430 A086584 A032014

Adjacent sequences: A002446 A002447 A002448 this_sequence A002450 A002451 A002452

nonn,nice,easy

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

Recurrence, Maple program and more terms from Michael Kleber (michael.kleber(AT)gmail.com), Dec 05 2000

Cayley reference from D. E. Knuth, Aug 17, 2001