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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

Table of n, a(n) for n=1..83.

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 November 13 00:25 EST 2019. Contains 329083 sequences. (Running on oeis4.)