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%I #12 Apr 21 2023 14:31:10
%S 1,1,2,4,11,31,106,372,1500,6220,28696,136016,702802,3727946,21253324,
%T 124231096,772458367,4918962479,33061095118,227303570524,
%U 1639389365201,12082068856285,92951844299150,729991789222972,5960617085224012,49637008114202876
%N Number of partitions of {1,...,n} into block sizes a power of 2.
%H Alois P. Heinz, <a href="/A115625/b115625.txt">Table of n, a(n) for n = 0..604</a>
%F E.g.f.: exp(x+x^2/2+x^4/4!+x^8/8!+x^16/16!+...).
%p with(combinat):
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p b(n-i*j, i/2), j=0..n/i)))
%p end:
%p a:= n-> b(n, 2^ilog2(n)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 17 2015
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add((j->
%p a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 21 2023
%t multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i/2], {j, 0, n/i}]]]; a[n_] := b[n, 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 29 2015, after _Alois P. Heinz_ *)
%Y Cf. A018819, A115626.
%K nonn
%O 0,3
%A _Christian G. Bower_, Jan 26 2006
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