A303064 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.

1, 1, 3, 19, 181, 2121, 28035, 401199, 6076494, 96043696, 1569407902, 26338955412, 451829668906, 7894022242204, 140075248932330, 2518908613782600, 45823514062061016, 842108582257569081, 15614889082228858722, 291858857158743005901, 5494258022591894716440, 104097462455681871262881, 1983820645046435115347970, 38007365345354099879246673

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (terms 0..200 from Paul D. Hanna)

Formula

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

(1) A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.

(2) A(x) = 1 + Integral A(x)^3 * (A(x) - 4*x*A'(x)) / (A(x) - 7*x*A'(x)) dx.

(3) A(x) = 1 + Integral A(x) * (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4) )/(14*x) dx.

(4) 0 = A(x)^4 - A(x)*(1 + 4*x*A(x)^2)*A'(x) + 7*x*A'(x)^2.

a(n) ~ c * d^n / n^(5/2), where d = 2^(5/3)*(3/7)^(4/9)*(432893979 + 94465211*sqrt(21))^(1/9) = 21.43962319881971664190505405921680468600... and c = 0.022570265358175200394042178896826753964244... - Vaclav Kotesovec, Oct 14 2020, updated Mar 16 2024

Example

G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 181*x^4 + 2121*x^5 + 28035*x^6 + 401199*x^7 + 6076494*x^8 + 96043696*x^9 + 1569407902*x^10 + ...
RELATED SERIES.
(x/A(x)^4)' / (x/A(x)^7)' = 1 + 6*x + 57*x^2 + 724*x^3 + 10605*x^4 + 168210*x^5 + 2808393*x^6 + 48611952*x^7 + 864393264*x^8 + 15694079020*x^9 + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 49*x^2 + 641*x^3 + 9541*x^4 + 152789*x^5 + 2567293*x^6 + 44643689*x^7 + 796602019*x^8 + 14502820745*x^9 + ...
which equals (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4))/(14*x).

Mathematica

nmax = 25; A = 1; Do[A = 1 + Integrate[D[x/A^4, x]/D[x/A^7, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] (* Vaclav Kotesovec, Oct 15 2020 *)

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A^4)'/(x/A^7 +x*O(x^n))' ); ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A302701, A302704, A302705, A338163, A338187, A338188, A338193, A338194.

Sequence in context: A269421 A304578 A306576 * A343672 A161630 A371320

Adjacent sequences: A303061 A303062 A303063 * A303065 A303066 A303067

Keyword

nonn

Author

Paul D. Hanna, Apr 19 2018

Status

approved