A370546 -id:A370546 - cp's OEIS frontend

A370439 Expansion of g.f. A(x) satisfying A(x) = A( xA(x)^2 + 3x*A(x)^3 )^(1/3).

1, 3, 9, 30, 126, 648, 3591, 19953, 110079, 610500, 3440493, 19742616, 114918138, 675417474, 3996992547, 23791052862, 142393544757, 856746349992, 5179722791274, 31449875426622, 191678795532801, 1172198278949454, 7190652243631437, 44235165115911312, 272837082264574914

Offset: 1

Paul D. Hanna, Mar 27 2024

nonn

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

			G.f.: A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 126*x^5 + 648*x^6 + 3591*x^7 + 19953*x^8 + 110079*x^9 + 610500*x^10 + 3440493*x^11 + 19742616*x^12 + ...
where A(x)^3 = A( x*A(x)^2 + 3*x*A(x)^3 ).

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A356782, A117940, A370546.

{a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^2 + 3*x*F^3) - F^3,#A+1) );A[n+1]}
for(n=1,30, print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1.a) A(x)^3 = A( x*A(x)^2 * (1 + 3*A(x)) ).
(1.b) A(x)^9 = A( x*A(x)^8 * (1 + 3*A(x))*(1 + 3*A(x)^3) ).
(1.c) A(x)^27 = A( x*A(x)^26 * (1 + 3*A(x))*(1 + 3*A(x)^3)*(1 + 3*A(x)^9) ).
(1.d) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + 3*A(x)^(3^k)) ).
(2) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(3^n)).
(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(3^n)) ).
(4) A(x) = x * Sum_{n>=0} A117940(n) * A(x)^n, where g.f. of A117940 equals Product{k>=0} 1 + 3*x^(3^k).
a(n) ~ c * d^n / n^(3/2), where d = 6.5583689184153129045048... and c = 0.129061736750222730297... - Vaclav Kotesovec, Apr 05 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^(n+1) * A(r)^(3^n) / (1 + 3*A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(3^n)), where r = 0.1524769363297159918479... = 1/d (d is given above) and A(r) = 0.3905308673397427979651361312666180120359942797557... - Paul D. Hanna, May 22 2024

A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

Original entry on oeis.org

1, 1, 4, 21, 125, 801, 5388, 37518, 268109, 1955000, 14487754, 108794169, 826054062, 6331064385, 48914088750, 380555960864, 2978892961194, 23444095375593, 185394136871818, 1472396312841250, 11739089289817538, 93921736129064325, 753845680317416682, 6068255413854119432

Offset: 1

Paul D. Hanna, Mar 26 2024

nonn

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

			G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
where A(x)^5 = A( x^5 + 5*A(x)^6 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
then B(x)^5 + 5*x^6 = B(x^5).
Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A352703, A091200, A271931, A370546.

{a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

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A370439 Expansion of g.f. A(x) satisfying A(x) = A( xA(x)^2 + 3x*A(x)^3 )^(1/3).

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A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

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cp's OEIS Frontend

A370439 Expansion of g.f. A(x) satisfying A(x) = A( x*A(x)^2 + 3*x*A(x)^3 )^(1/3).

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A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

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A370439 Expansion of g.f. A(x) satisfying A(x) = A( xA(x)^2 + 3x*A(x)^3 )^(1/3).