A376175 - OEIS

%I #17 Sep 15 2024 08:02:08

%S 1,1,7,97,2035,57445,2042215,87651865,4410770875,254705483725,

%T 16603869256975,1206175463317825,96627476254984675,

%U 8463175473211383925,804573717383525464375,82513390092813146091625,9080444173122231239204875,1067360792025339122846660125,133468583774114314367364097375

%N E.g.f. A(x) satisfies: A'(x) = 1 + A(x)*A'(x)^3.

%H Paul D. Hanna, <a href="/A376175/b376175.txt">Table of n, a(n) for n = 1..300</a>

%F E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies:

%F (1) A(x) = Series_Reversion( Integral( dx/D(x) ) ) where D(x) = 1 + x*D(x)^3 is the g.f. of A001764.

%F (2) A'(x) = Sum_{n>=0} A001764(n)*A(x)^n where A001764(n) = binomial(3*n,n)/(2*n+1).

%F (3) A'(x) = 1 + A(x)*A'(x)^3.

%F (4) A'(x) = 1/(1 - A(x)*A'(x)^2).

%F (5) A(x) = x + Sum_{n>=0} d^n/dx^n G(x)^(n+1)/(n+1)!, where G(x) = Integral( x*D(x)^2 dx ) and D(x) = 1 + x*D(x)^3.

%F (6) A(x - B(x)) = x where B(x) = Sum_{n>=0} binomial(3*n+1,n) * x^(n+2) / ((n+1)*(n+2)).

%F a(n) ~ 2^(2*n - 6) * 3^(4*n - 7/2) * n^(n-2) / (exp(n) * 43^(n - 3/2)). - _Vaclav Kotesovec_, Sep 14 2024

%e E.g.f.: A(x) = x + x^2/2! + 7*x^3/3! + 97*x^4/4! + 2035*x^5/5! + 57445*x^6/6! + 2042215*x^7/7! + 87651865*x^8/8! + 4410770875*x^9/9! + ...

%e where A'(x) = 1 + A(x)*A'(x)^3.

%e Also,

%e A'(x) = 1 + A(x) + 3*A(x)^2 + 12*A(x)^3 + 55*A(x)^4 + 273*A(x)^5 + 1428*A(x)^6 + 7752*A(x)^7 + ... + A001764(n)*A(x)^n + ...

%e RELATED SERIES.

%e Series D(x) = 1 + x*D(x)^3 begins

%e D(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + ... + A001764(n)*x^n + ...

%e where Integral( dx/D(x) ) = x - x^2/2! - 4*x^3/3! - 42*x^4/4! - 720*x^5/5! - 17160*x^6/6! - 524160*x^7/7! - 19535040*x^8/8! + ...

%e and A(x) = Series_Reversion( Integral( dx/D(x) ) ).

%e A'(x)^3 = 1 + 3*x + 27*x^2/2! + 423*x^3/3! + 9567*x^4/4! + 284355*x^5/5! + 10499715*x^6/6! + 464006655*x^7/7! + ...

%o (PARI) /* A'(x) = 1 + A(x)*A'(x)^3 */

%o {a(n) = my(A=x); for(i=1, n, A = intformal(1 + A*(A')^3 + x*O(x^n))); n!*polcoeff(A, n)}

%o for(n=1, 20, print1(a(n), ", "))	

%o (PARI) /* A(x) = Series_Reversion( Integral( dx/D ) ) where D = 1 + x*D^3 */

%o {a(n) = my(D = ((1/x)*serreverse(x/(1 + x + x*O(x^n))^3))^(1/3), A); A = serreverse(intformal(1/D)); n!*polcoeff(A, n)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A180254, A376174, A001764.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 13 2024