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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. 5
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS 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} REFERENCES G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169. LINKS FORMULA p(k, x) = Sum[p(j, x)*p(k - j, x), {j, 2, k - 1}]. EXAMPLE Triangular sequence {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, 35, 0, 70, 0, 21, 0, 1}, {14, 0, 140, 0, 140, 0, 28, 0, 1} MAPLE 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 MATHEMATICA 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] CROSSREFS See A097610 for another version. Cf. A072851. Sequence in context: A199176 A021336 A100749 * A097610 A161556 A317302 Adjacent sequences:  A124024 A124025 A124026 * A124028 A124029 A124030 KEYWORD nonn,tabf,easy AUTHOR Roger L. Bagula, Oct 31 2006 EXTENSIONS Edited by N. J. A. Sloane, Oct 07 2007 More terms from R. J. Mathar, Oct 08 2007 STATUS approved

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Last modified April 6 11:38 EDT 2020. Contains 333273 sequences. (Running on oeis4.)