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A182135 Number of partitions of 2^2^n into powers of 2 less than or equal to 2^n. 3
1, 3, 25, 47905, 751333186150401, 371679100488302192208527928207947545444353 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Lengths (in decimal digits) of the terms a(0), a(1), ... are: 1, 1, 2, 5, 15, 42, 107, 258, 602, 1369, 3060, 6755, 14765, 32022, 69007, 147915, 315599, 670702, 1420371, ... .

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..8

FORMULA

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

EXAMPLE

a(1) = 3 because the number of partitions of 2^2^1 = 4 into powers of 2 less than or equal to 2^1 = 2 is 3: [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(2^(2^n-n), n):

seq(a(n), n=0..10);

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[2^(2^n-n), n]; Table[a[n], {n, 0, 5}] (* Jean-Fran├žois Alcover, Feb 05 2017, translated from Maple *)

CROSSREFS

Main diagonal of A172288.

Sequence in context: A246536 A183248 A144788 * A307654 A307653 A326610

Adjacent sequences:  A182132 A182133 A182134 * A182136 A182137 A182138

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 26 2012

STATUS

approved

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Last modified May 31 02:51 EDT 2020. Contains 334747 sequences. (Running on oeis4.)