

A138056


Levels of substitution A103684 (based on the morphism f: 1>{1,2}, 2>{1,3}, 3>{3}) like Markov substitution taken as polynomials p(x,n)]and coefficients of the differential polynomials returned as q(x,n) =dp(x,n)dx coefficients. ( first zero omitted).


0



2, 2, 2, 9, 2, 2, 9, 4, 10, 6, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 24, 50, 26, 81, 28, 58, 30, 31, 64, 33, 102, 35
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OFFSET

1,1


COMMENTS

Row sums with zero: {0, 2, 13, 33, 130, 459, 1533, 5266, 17884, 60532, 205129, ...};
This sequence uses the Markov substitution form that I have been using in my chordgeometry/ graph sequences.
This method of differentiating a substitution appears to be new.


LINKS

Table of n, a(n) for n=1..80.


FORMULA

f: 1>{1,2}, 2>{1,3}, 3>{3}); p(x,n)=Sum[Substitution[n,m]*t(m1),{m,1,n}]; q(x,n)=dp(x,n)dx; out_n,m=Coefficients(q(x,n).


EXAMPLE

{2},
{2, 2, 9},
{2, 2, 9, 4, 10, 6},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69},
{2, 2, 9, 4, 10, 6, 7, 16, 9, 30, 11, 24, 13, 28, 15, 48, 17, 36, 19, 20, 42, 22, 69, 24, 50, 26, 81, 28, 58, 30, 31, 64, 33, 102, 35, 72, 37, 76, 39, 120, 41, 84, 43}


MATHEMATICA

Clear[a, s, p, t, m, n]; (* substitution *); s[1] = {1, 2}; s[2] = {1, 3}; s[3] = {1}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n  1]]; (*A103684*); a = Table[p[n], {n, 0, 10}]; Flatten[a]; b = Table[CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]*x^( m  1), {m, 1, Length[a[[n]]]}]], x], x], {n, 1, 11}]; Flatten[b] Table[Apply[Plus, CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]* x^(m  1), {m, 1, Length[a[[n]]]}]], x], x]], {n, 1, 11}];


CROSSREFS

Cf. A103684.
Sequence in context: A298647 A068718 A075097 * A340080 A022459 A060804
Adjacent sequences: A138053 A138054 A138055 * A138057 A138058 A138059


KEYWORD

nonn,uned,tabf


AUTHOR

Roger L. Bagula, May 02 2008


STATUS

approved



