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A124027 G. J. Chaitin's numbers of s-expressions of size n are given by the coefficients of polynomials p(k, x) satisfying p(k, x) = Sum[p(j, x)*p(k - j, x), {j, 2, k - 1}]. The coefficients of these polynomials give the triangle shown here.

%I A124027 #11 Aug 12 2015 02:12:38
%S A124027 0,0,1,1,0,1,1,0,1,0,3,0,1,2,0,6,0,1,0,10,0,10,0,1,5,0,30,0,15,0,1,0,
%T A124027 35,0,70,0,21,0,1,14,0,140,0,140,0,28,0,1,0,126,0,420,0,252,0,36,0,1,
%U A124027 42,0,630,0,1050,0,420,0,45,0,1,0,462,0,2310,0,2310,0,660,0,55,0,1,132,0
%N A124027 G. J. Chaitin's numbers of s-expressions of size n are given by the coefficients of polynomials p(k, x) satisfying p(k, x) = Sum[p(j, x)*p(k - j, x), {j, 2, k - 1}]. The coefficients of these polynomials give the triangle shown here.
%C A124027 Row sum sequence: Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[n, x], x]]}], {n, 0, 15}] {0, 1, 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835}
%D A124027 G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169.
%F A124027 p(k, x) = Sum[p(j, x)*p(k - j, x), {j, 2, k - 1}].
%e A124027 Triangular sequence
%e A124027 {0},
%e A124027 {0, 1},
%e A124027 {1},
%e A124027 {0, 1},
%e A124027 {1, 0, 1},
%e A124027 {0, 3, 0, 1},
%e A124027 {2, 0, 6, 0, 1},
%e A124027 {0, 10, 0, 10, 0, 1},
%e A124027 {5, 0, 30, 0, 15, 0, 1},
%e A124027 {0, 35, 0, 70, 0, 21, 0, 1},
%e A124027 {14, 0, 140, 0, 140, 0, 28, 0, 1}
%p A124027 p := proc(k,x) option remember ; if k = 0 then 0 ; elif k= 1 then x; elif k= 2 then 1; else add(p(j,x)*p(k-j,x),j=2..k-1) ; fi ; end: A124027 := proc(n,k) coeftayl( p(n,x),x=0,k) ; end: printf("0, 0, 1, ") ; for n from 0 to 18 do for k from 0 to n-2 do printf("%d, ",A124027(n,k)) ; od: od: # _R. J. Mathar_, Oct 08 2007
%t A124027 p[0, x] = 0; p[1, x] = x; p[2, x] = 1; p[k_, x_] := p[k, x] = Sum[p[j, x]*p[k - j, x], {j, 2, k - 1}]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
%Y A124027 See A097610 for another version. Cf. A072851.
%K A124027 nonn,tabf,easy
%O A124027 1,11
%A A124027 _Roger L. Bagula_, Oct 31 2006
%E A124027 Edited by _N. J. A. Sloane_, Oct 07 2007
%E A124027 More terms from _R. J. Mathar_, Oct 08 2007