A268294 E.g.f. satisfies: A(x) = Sum_{n>=1} [Integral exp(n*A(x)) dx]^n/n.

1, 2, 11, 104, 1437, 26642, 629127, 18272434, 636604869, 26108399430, 1242008775951, 67720430926014, 4190308707363309, 291752805954748278, 22688876961654354279, 1957921721567050433622, 186386540343460036556349, 19470192489522569878618398, 2221126565909171872353071895, 275498553541064512468727996046, 37005623218402026792412910411205

Offset

1,2

Comments

Compare to: if G(x) = Sum_{n>=1} [Integral exp((t-1)*G(x)) dx]^n/n then exp(t*G(x)) = 1/(1-t*x).

Links

Table of n, a(n) for n=1..21.

Formula

E.g.f. satisfies: A'(x) = Sum_{n>=1} exp(n*A(x)) * [Integral exp(n*A(x)) dx]^(n-1).

Example

E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1437*x^5/5! + 26642*x^6/6! + 629127*x^7/7! + 18272434*x^8/8! + 636604869*x^9/9! + 26108399430*x^10/10! +...
Let B(n,x) = Integral exp(n*A(x)) dx and C(x) = exp(A(x)), then
A(x) = B(1,x) + B(2,x)^2/2 + B(3,x)^3/3 + B(4,x)^4/4 + B(5,x)^5/5 + B(6,x)^6/6 + B(7,x)^7/7 + B(8,x)^8/8 +...
A'(x) = C(x) + C(x)^2*B(2,x) + C(x)^3*B(3,x)^2 + C(x)^4*B(4,x)^3 + C(x)^5*B(5,x)^4 + C(x)^6*B(6,x)^5 + C(x)^7*B(7,x)^6 + C(x)^8*B(8,x)^7 +...
The coefficients of x^k/k! in B(n,x), k>=1, begin:
n=1: [1, 1, 3, 18, 173, 2368, 43025, 991070, 28030227 ...];
n=2: [1, 2, 8, 54, 544, 7546, 136516, 3100838, 86014692 ...];
n=3: [1, 3, 15, 114, 1221, 17424, 317913, 7192782, 197168727 ...];
n=4: [1, 4, 24, 204, 2336, 34612, 643016, 14615900, 398769096 ...];
n=5: [1, 5, 35, 330, 4045, 62560, 1191385, 27393710, 748596795 ...];
n=6: [1, 6, 48, 498, 6528, 105678, 2072460, 48458610, 1333466412 ...];
n=7: [1, 7, 63, 714, 9989, 169456, 3432401, 81936638, 2280265407 ...];
n=8: [1, 8, 80, 984, 14656, 260584, 5461648, 133487672, 3769966992, ...]; ...
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 18*x^3/3! + 173*x^4/4! + 2368*x^5/5! + 43025*x^6/6! + 991070*x^7/7! + 28030227*x^8/8! + 950818494*x^9/9! + 37995695979*x^10/10! +...

Mathematica

a[n_] := (A = x + x*O[x]^n; For[i = 1, i <= n + 1, i++, A = Sum[Integrate[Exp[m*A], x]^m/m, {m, 1, n}]]; n!*SeriesCoefficient[A, {x, 0, n}]); Array[a, 21] (* Jean-François Alcover, Oct 03 2016, adapted from PARI *)

Prog

(PARI) {a(n) = my(A=x +x*O(x^n)); for(i=1, n+1, A = sum(m=1, n, intformal(exp(m*A))^m/m ) ); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A232552.

Sequence in context: A277470 A343896 A111535 * A364338 A207571 A351331

Adjacent sequences: A268291 A268292 A268293 * A268295 A268296 A268297

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2016

Status

approved