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A036718
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Number of rooted trees where each node has at most 4 children.
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13
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1, 1, 1, 2, 4, 9, 19, 45, 106, 260, 643, 1624, 4138, 10683, 27790, 72917, 192548, 511624, 1366424, 3666930, 9881527, 26730495, 72556208, 197562840, 539479354, 1477016717, 4053631757, 11149957667, 30732671572, 84871652538, 234802661446, 650684226827
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listen;
history;
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f. satisfies A(x) = 1 + x*cycle_index(Sym(4), A(x)).
a(n) / a(n+1) ~ 0.343520104570489046632074698738792654644751898257681287407149... - Robert A. Russell, Feb 11 2023
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EXAMPLE
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The a(5) = 9 rooted trees with 5 nodes and out-degrees <= 4 are:
: level sequence out-degrees (dots for zeros)
: 1: [ 0 1 2 3 4 ] [ 1 1 1 1 . ]
: O--o--o--o--o
:
: 2: [ 0 1 2 3 3 ] [ 1 1 2 . . ]
: O--o--o--o
: .--o
:
: 3: [ 0 1 2 3 2 ] [ 1 2 1 . . ]
: O--o--o--o
: .--o
:
: 4: [ 0 1 2 3 1 ] [ 2 1 1 . . ]
: O--o--o--o
: .--o
:
: 5: [ 0 1 2 2 2 ] [ 1 3 . . . ]
: O--o--o
: .--o
: .--o
:
: 6: [ 0 1 2 2 1 ] [ 2 2 . . . ]
: O--o--o
: .--o
: .--o
:
: 7: [ 0 1 2 1 2 ] [ 2 1 . 1 . ]
: O--o--o
: .--o--o
:
: 8: [ 0 1 2 1 1 ] [ 3 1 . . . ]
: O--o--o
: .--o
: .--o
:
: 9: [ 0 1 1 1 1 ] [ 4 . . . . ]
: O--o
: .--o
: .--o
: .--o
(End)
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MAPLE
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A := 1; f := proc(n) global A; local A2, A3, A4; A2 := subs(x=x^2, A); A3 := subs(x=x^3, A); A4 := subs(x=x^4, A);
coeff(series( 1+x*( (A^4+3*A2^2+8*A*A3+6*A^2*A2+6*A4)/2 ), x, n+1), x, n); end;
for n from 1 to 50 do A := series(A+f(n)*x^n, x, n +1); od: A;
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MATHEMATICA
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a = 1; f[n_] := Module[{a2, a3, a4}, a2 = a /. x -> x^2; a3 = a /. x -> x^3; a4 = a /. x -> x^4; Coefficient[ Series[ 1 + x*(a^4 + 3*a2^2 + 8*a*a3 + 6*a^2*a2 + 6*a4)/24, {x, 0, n + 1}] // Normal, x, n]]; For[n = 1, n <= 30, n++, a = Series[a + f[n]*x^n, {x, 0, n + 1}] // Normal]; CoefficientList[a, x] (* Jean-François Alcover, Jan 16 2013, after Maple *)
b[0, i_, t_, k_] = 1; m = 4; (* m = maximum children *)
b[n_, i_, t_, k_]:= b[n, i, t, k]= If[i<1, 0,
Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
PrependTo[Table[b[n-1, n-1, m, m], {n, 1, 30}], 1] (* Robert A. Russell, Dec 27 2022 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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