A002577 - OEIS

A002577

Number of partitions of 2^n into powers of 2.
(Formerly M1239 N0473)

1, 2, 4, 10, 36, 202, 1828, 27338, 692004, 30251722, 2320518948, 316359580362, 77477180493604, 34394869942983370, 27893897106768940836, 41603705003444309596874, 114788185359199234852802340, 588880400923055731115178072778, 5642645813427132737155703265972004

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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 rows IV, V etc. of the given table are not represented in the OEIS till now. - Valentin Bakoev, Feb 25 2009; edited by M. F. Hasler, Feb 09 2014

REFERENCES

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.

Lawrence, Jim. "Dual-Antiprisms and Partitions of Powers of 2 into Powers of 2." Discrete & Computational Geometry, Vol. 16 (2019): 465-478. See page 466.

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).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..85

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

C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, 2012. - From N. J. A. Sloane, Dec 23 2012

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

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

R. F. Churchhouse, Congruence properties of the binary partition function, Math. Proc. Cambr. Phil. Soc. vol 66, no. 2 (1969), 365-370.

Carl-Erik Froberg, Accurate estimation of the number of binary partitions, BIT Numerical Mathematics vol. 17, no 4 (1977) 386-391.

C.-E. Froberg, Accurate estimation of the number of binary partitions [Annotated scanned copy]

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

FORMULA

a(n) is about 0.9233*Sum_j {i=0, 1, 2, 3, ...} 2^(j*(2n-j-1)/2)/j!. - Henry Bottomley, Jul 23 2003

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

A002577(n)-1 = A125792(n). - Let m > 1, n > 0 and k >= 0. The general formula for the number of all partitions of 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,... - Valentin Bakoev, Feb 25 2009

a(n) = [x^(2^n)] 1/Product_{j>=0} (1-x^(2^j)). - Alois P. Heinz, Sep 27 2011

EXAMPLE

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. - Valentin Bakoev, Feb 25 2009

G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 36*x^4 + 202*x^5 + 1828*x^6 + ...

MAPLE

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

MATHEMATICA

$RecursionLimit = 10^5; (* b = A000123 *) b[0] = 1; b[n_?EvenQ] := b[n] = b[n-1] + b[n/2]; b[n_?OddQ] := b[n] = b[n-1] + b[(n-1)/2]; a[n_] := b[2^(n-1)]; a[0] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 23 2011 *)

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^2^k, {k, 0, n}], {x, 0, 2^n}]; (* Michael Somos, Apr 21 2014 *)

PROG

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

(Haskell)

import Data.MemoCombinators (memo2, list, integral)

a002577 n = a002577_list !! n

a002577_list = f [1] where

f xs = (p' xs $ last xs) : f (1 : map (* 2) xs)

p' = memo2 (list integral) integral p

p _ 0 = 1; p [] _ = 0

p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m

-- Reinhard Zumkeller, Nov 27 2015

CROSSREFS

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

Cf. A078537, A078121, A000290, A000447.

Column k=2 of A145515, diagonal of A152977. - Alois P. Heinz, Mar 25 2012