A212775 Number of partitions of 2^(2^n) into powers of 2.

2, 4, 36, 692004, 114788185359199234852802340

Offset

0,1

Comments

Lengths (in decimal digits) of the terms a(0), a(1), ... are: 1, 1, 2, 6, 27, 119, 525, 2241, 9330, ... .

Links

Alois P. Heinz, Table of n, a(n) for n = 0..6

Formula

a(n) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..2^n-1} (1-x^(2^j)).

Example

a(0) = 2 because the number of partitions of 2^2^0 = 2 into powers of 2 is 2: [2], [1,1].
a(1) = 4: [4], [2,2], [2,1,1], [1,1,1,1].

Maple

b:= proc(n, j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n<j then b(n, j):= b(n-1, j) +b(2*n, j-1)
             else nn:= 1 +floor(n);
                  r:= n-nn;
                  (nn-j) *binomial(nn, j) *add(binomial(j, h)
                  /(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
      fi
    end:
a:= n-> b(1, 2^n):
seq(a(n), n=0..6);

Mathematica

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j==0, 1, j==1, n+1, n < j, b[n, j] = b[n-1, j] + b[2*n, j-1], True, nn = 1+Floor[n]; r = n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_] := b[1, 2^n]; Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

Crossrefs

Cf. A000123, A002577, A172288, A182135.

Sequence in context: A326932 A063184 A323717 * A115250 A018304 A289988

Adjacent sequences: A212772 A212773 A212774 * A212776 A212777 A212778

Keyword

nonn

Author

Alois P. Heinz, May 26 2012

Status

approved