A280574 E.g.f. satisfies: A(x - Integral 3*A(x) dx) = x + Integral 2*A(x) dx.

1, 5, 70, 1775, 66175, 3283475, 204594175, 15411893450, 1366394303500, 139767921720875, 16243630181913625, 2118892887756520250, 307173379745256857875, 49084564051462443496250, 8586127214178418541668750, 1634509914502001105016284375, 336910750825106071274158853125, 74862327518834451026921878887500, 17862833297180486514281227128971875, 4561279298680105599840369905594562500

Offset

1,2

Links

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

Formula

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

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

(2) A(x) = x + 5 * G( (3*A(x) + 2*x)/5 ), where G(x) = Integral A(x) dx.

(3) A(x) = -2*x/3 + 5/3 * Series_Reversion(x - Integral 3*A(x) dx).

(4) A( (3*A(x) + 2*x)/5 ) = (A'(x) - 1)/(3*A'(x) + 2).

(5) A'(x - Integral 3*A(x) dx) = (1 + 2*A(x))/(1 - 3*A(x)).

a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 5^(n-k-1).

Example

E.g.f.: A(x) = x + 5*x^2/2! + 70*x^3/3! + 1775*x^4/4! + 66175*x^5/5! + 3283475*x^6/6! + 204594175*x^7/7! + 15411893450*x^8/8! + 1366394303500*x^9/9! + 139767921720875*x^10/10! + 16243630181913625*x^11/11! + 2118892887756520250*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 3*G(x)) = x + 2*G(x) where
G(x) = x^2/2! + 5*x^3/3! + 70*x^4/4! + 1775*x^5/5! + 66175*x^6/6! + 3283475*x^7/7! + 204594175*x^8/8! + 15411893450*x^9/9! + 1366394303500*x^10/10! + 139767921720875*x^11/11! + 16243630181913625*x^12/12! +...
Also, A(x) = x + 5 * G( (3*A(x) + 2*x)/5 ).
RELATED SERIES.
We have (3*A(x) + 2*x)/5 = Series_Reversion( x - Integral 3*A(x) dx ), where
(3*A(x) + 2*x)/5 = x + 3*x^2/2! + 42*x^3/3! + 1065*x^4/4! + 39705*x^5/5! + 1970085*x^6/6! + 122756505*x^7/7! + 9247136070*x^8/8! + 819836582100*x^9/9! + 83860753032525*x^10/10! + 9746178109148175*x^11/11! + 1271335732653912150*x^12/12! +...
Further, A( (3*A(x) + 2*x)/5 ) = (A'(x) - 1)/(3*A'(x) + 2), which begins
A( (3*A(x) + 2*x)/5 ) = x + 8*x^2/2! + 157*x^3/3! + 5075*x^4/4! + 230905*x^5/5! + 13636085*x^6/6! + 994743280*x^7/7! + 86697077570*x^8/8! + 8813260716925*x^9/9! + 1026216275720525*x^10/10! + 134948279040712300*x^11/11! + 19814992125974741525*x^12/12! +...

Mathematica

m = 21; A[_] = 0;
Do[A[x_] = -2x/3 + 5/3 InverseSeries[x-Integrate[3A[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=3, q=2) = 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, 3, 2), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=3, q=2) = 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, 3, 2), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=3; q=2; 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, A280575.

Sequence in context: A203528 A147629 A274256 * A302910 A370576 A174486

Adjacent sequences: A280571 A280572 A280573 * A280575 A280576 A280577

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2017

Status

approved