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Number of compositions of 2^n into distinct parts.
0

%I #16 Oct 05 2015 11:20:58

%S 1,1,3,19,435,74875,348317763,294729601581739,

%T 682404222981720262704195,298417646219775679438413815505895285915,

%U 13661663328896434876017827688479176004409461863714010289523203

%N Number of compositions of 2^n into distinct parts.

%H Henry Bottomley, <a href="http://www.se16.info/js/partitionstest.htm">Partition and composition calculator using a Java applet</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F a(n) = A032020(A000079(n)).

%e a(2) = 3 since the compositions of 2^2=4 into distinct parts are 4, 3+1 and 1+3.

%p b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;

%p `if`(n=m, x^i, `if`(n>m, 0,

%p expand(b(n, i-1)+`if`(i>n, 0, x*b(n-i, i-1)))))

%p end:

%p a:= n->(p->add(coeff(p, x, i)*i!, i=0..degree(p)))(b(2^n$2)):

%p seq(a(n), n=0..9); # _Alois P. Heinz_, Apr 27 2014

%t b[n_, i_] := b[n, i] = With[{ m = i*(i+1)/2}, If[n==m, x^i, If[n>m, 0, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*i!, {i, 0, Exponent[p, x]}]][b[2^n, 2^n]]; Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Oct 05 2015, after _Alois P. Heinz_ *)

%Y Cf. A058891 (offset for compositions of 2^n), A067735, A068413.

%K nonn

%O 0,3

%A _Henry Bottomley_, Feb 08 2003