A280570 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

1, 4, 28, 332, 5748, 131940, 3791692, 131375324, 5343640212, 250142552212, 13271217848604, 788346022938556, 51916178572447140, 3759254932421361284, 297243198474965188732, 25513664852425377663756, 2365246919693613357168916, 235776253411115081902083556, 25174157913006507920211300588, 2869108641038261410331666767772

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

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

(1) A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

(2) A(x) = x + 4 * G( (A(x) + 3*x)/4 ), where G(x) = Integral A(x) dx.

(3) A(x) = -3*x + 4 * Series_Reversion(x - Integral A(x) dx).

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

(5) A'(x - Integral A(x) dx) = (1 + 3*A(x))/(1 - A(x)).

a(n) = Sum_{k=0..n} A277410(n,k) * 4^(n-k-1).

Example

E.g.f.: A(x) = x + 4*x^2/2! + 28*x^3/3! + 332*x^4/4! + 5748*x^5/5! + 131940*x^6/6! + 3791692*x^7/7! + 131375324*x^8/8! + 5343640212*x^9/9! + 250142552212*x^10/10! + 13271217848604*x^11/11! + 788346022938556*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 3*G(x) where
G(x) = x^2/2! + 4*x^3/3! + 28*x^4/4! + 332*x^5/5! + 5748*x^6/6! + 131940*x^7/7! + 3791692*x^8/8! + 131375324*x^9/9! + 5343640212*x^10/10! + 250142552212*x^11/11! + 13271217848604*x^12/12! +...
Also, A(x) = x + 4 * G( (A(x) + 3*x)/4 ).
RELATED SERIES.
We have (A(x) + 3*x)/4 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 3*x)/4 = x + x^2/2! + 7*x^3/3! + 83*x^4/4! + 1437*x^5/5! + 32985*x^6/6! + 947923*x^7/7! + 32843831*x^8/8! + 1335910053*x^9/9! + 62535638053*x^10/10! + 3317804462151*x^11/11! + 197086505734639*x^12/12! +...
Further, A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3), which begins
A( (A(x) + 3*x)/4 ) = x + 5*x^2/2! + 47*x^3/3! + 707*x^4/4! + 14825*x^5/5! + 401033*x^6/6! + 13340739*x^7/7! + 528281555*x^8/8! + 24323141773*x^9/9! + 1279128727141*x^10/10! + 75770789421947*x^11/11! + 4999463984999615*x^12 +...

Mathematica

m = 21; A[_] = 0;
Do[A[x_] = -3 x + 4 InverseSeries[x - Integrate[A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x] * Range[0, m-1]! // Rest (* Jean-François Alcover, Sep 30 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=1, q=3) = 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, 1, 3), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=1, q=3) = 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, 1, 3), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=1; q=3; 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, A279844, A279845, A280571, A280572, A280573, A280574, A280575.

Sequence in context: A358072 A198513 A113371 * A080636 A223887 A140425

Adjacent sequences: A280567 A280568 A280569 * A280571 A280572 A280573

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2017

Status

approved