A279844 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.

1, 3, 27, 441, 10593, 338715, 13603923, 660689217, 37773985257, 2492351980659, 186888829248171, 15733456044557193, 1472423968474987185, 151932311464679521803, 17166519680224611739203, 2111435499783771418877073, 281279117575497421255121721, 40406056752677361995435879907, 6234806360224720540046684747547, 1029860015641146082486445487150681

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..150

Formula

E.g.f. A(x) satisfies:

(1) A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.

(2) A(x) = x + 3 * G( (2*A(x) + x)/3 ), where G(x) = Integral A(x) dx.

(3) A(x) = -x/2 + 3/2 * Series_Reversion(x - Integral 2*A(x) dx).

(4) A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1).

(5) A'(x - Integral 2*A(x) dx) = (1 + A(x))/(1 - 2*A(x)).

a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 3^(n-k-1).

Example

E.g.f.: A(x) = x + 3*x^2/2! + 27*x^3/3! + 441*x^4/4! + 10593*x^5/5! + 338715*x^6/6! + 13603923*x^7/7! + 660689217*x^8/8! + 37773985257*x^9/9! + 2492351980659*x^10/10! + 186888829248171*x^11/11! + 15733456044557193*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 2*G(x)) = x + G(x) where
G(x) = x^2/2! + 3*x^3/3! + 27*x^4/4! + 441*x^5/5! + 10593*x^6/6! + 338715*x^7/7! + 13603923*x^8/8! + 660689217*x^9/9! + 37773985257*x^10/10! +...
Also, A(x) = x + 3 * G( (2*A(x) + x)/3 ).
RELATED SERIES.
We have (2*A(x) + x)/3 = Series_Reversion( x - Integral 2*A(x) dx ), where
(2*A(x) + x)/3 = x + 2*x^2/2! + 18*x^3/3! + 294*x^4/4! + 7062*x^5/5! + 225810*x^6/6! + 9069282*x^7/7! + 440459478*x^8/8! + 25182656838*x^9/9! + 1661567987106*x^10/10! + 124592552832114*x^11/11! + 10488970696371462*x^12/12! +...
Further, A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1), which begins
A( (2*A(x) + x)/3 ) = x + 5*x^2/2! + 63*x^3/3! + 1311*x^4/4! + 38445*x^5/5! + 1464381*x^6/6! + 68939271*x^7/7! + 3879180855*x^8/8! + 254691006453*x^9/9! + 19160241768837*x^10/10! + 1628342402620383*x^11/11! + 154564849209408975*x^12/12! +...

Mathematica

m = 21; A[_] = 0;
Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 3 G[(2 A[x] + x)/3] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! // Rest (* Jean-François Alcover, Oct 20 2019 *)

Prog

(PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=2, q=1) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 2, 1), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=2, q=1) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 2, 1), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=2; q=1; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Sequence in context: A221624 A108525 A136719 * A159600 A159601 A193541

Adjacent sequences: A279841 A279842 A279843 * A279845 A279846 A279847

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2016

Status

approved