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%I #42 Apr 06 2020 05:38:06
%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,2,1,1,
%T 1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,2,2,1,1,1,1,
%U 1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,2,1,1,1,1,1
%N Run Length Transform of S(n) = wt(n) = 0,1,1,2,1,2,2,3,1,... (cf. A000120).
%C The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).
%H Reinhard Zumkeller, <a href="/A246588/b246588.txt">Table of n, a(n) for n = 0..10000</a>
%p A000120 := proc(n) local w, m, i; w := 0; m :=n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;
%p ans:=[];
%p for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0;
%p for i from 1 to L1 do
%p if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
%p elif out1 = 0 and t1[i] = 1 then c:=c+1;
%p elif out1 = 1 and t1[i] = 0 then c:=c;
%p elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0;
%p fi;
%p if i = L1 and c>0 then lis:=[c, op(lis)]; fi;
%p od:
%p a:=mul(wt(i), i in lis);
%p ans:=[op(ans), a];
%p od:
%p ans;
%t f[n_] := DigitCount[n, 2, 1]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* _Jean-François Alcover_, Jul 11 2017 *)
%o (Haskell)
%o import Data.List (group)
%o a246588 = product . map (a000120 . length) .
%o filter ((== 1) . head) . group . a030308_row
%o -- _Reinhard Zumkeller_, Feb 13 2015, Sep 05 2014
%o (Python)
%o from operator import mul
%o from functools import reduce
%o from re import split
%o def A246588(n):
%o return reduce(mul,(bin(len(d)).count('1') for d in split('0+',bin(n)[2:]) if d)) if n > 0 else 1 # _Chai Wah Wu_, Sep 07 2014
%o (Sage) # uses[RLT from A246660]
%o A246588_list = lambda len: RLT(lambda n: sum(Integer(n).digits(2)), len)
%o A246588_list(88) # _Peter Luschny_, Sep 07 2014
%Y Cf. A000120.
%Y Cf. A167489, A030308.
%Y Run Length Transforms of other sequences: A071053, A227349, A246595, A246596, A246660, A246661, A246674.
%K nonn
%O 0,8
%A _N. J. A. Sloane_, Sep 05 2014
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