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A182135 Number of partitions of 2^2^n into powers of 2 less than or equal to 2^n.

%I A182135 #63 Sep 21 2018 22:42:09
%S A182135 1,3,25,47905,751333186150401,
%T A182135 371679100488302192208527928207947545444353
%N A182135 Number of partitions of 2^2^n into powers of 2 less than or equal to 2^n.
%C A182135 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, ... .
%H A182135 Alois P. Heinz, <a href="/A182135/b182135.txt">Table of n, a(n) for n = 0..8</a>
%F A182135 a(n) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..n-1} (1-x^(2^j)).
%e A182135 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].
%p A182135 b:= proc(n, j) option remember; local nn, r;
%p A182135       if n<0 then 0
%p A182135     elif j=0 then 1
%p A182135     elif j=1 then n+1
%p A182135     elif n<j then b(n, j):= b(n-1, j) +b(2*n, j-1)
%p A182135              else nn:= 1 +floor(n);
%p A182135                   r:= n-nn;
%p A182135                   (nn-j) *binomial(nn, j) *add(binomial(j, h)
%p A182135                   /(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
%p A182135       fi
%p A182135     end:
%p A182135 a:= n-> b(2^(2^n-n), n):
%p A182135 seq(a(n), n=0..10);
%t A182135 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-François Alcover_, Feb 05 2017, translated from Maple *)
%Y A182135 Main diagonal of A172288.
%K A182135 nonn
%O A182135 0,2
%A A182135 _Alois P. Heinz_, May 26 2012