A232552 E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(x)^n dx )^n/n!, where the constant of integration is zero.

1, 1, 2, 9, 64, 630, 8030, 126247, 2371612, 52026293, 1309661828, 37318672196, 1190672836246, 42159850045181, 1644546319080848, 70233352188006641, 3266637689293293616, 164720219739258021686, 8969422973088951968070, 525585300443124229026511, 33039986976855724686082476

Offset

0,3

Comments

Compare to the identity:

if G(x) = Sum_{n>=0} Integral( G(x)^t dx )^n/n!, then G(x)^t = 1/(1 - t*x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Formula

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

Example

G.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 630*x^5/5! + 8030*x^6/6! +...
Let B(n,x) = Integral( A(x)^n dx ) with B(n,0)=0, then
A(x) = 1 + B(1,x) + B(2,x)^2/2! + B(3,x)^3/3! + B(4,x)^4/4! + B(5,x)^5/5! +...
A'(x) = A(x) + A(x)^2*B(2,x) + A(x)^3*B(3,x)^2/2! + A(x)^4*B(4,x)^3/3! +...
where
B(1,x) = x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 64*x^5/5! + 630*x^6/6! +...
B(2,x) = x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 224*x^5/5! + 2260*x^6/6! +...
B(3,x) = x + 3*x^2/2! + 12*x^3/3! + 69*x^4/4! + 552*x^5/5! + 5790*x^6/6! +...
B(4,x) = x + 4*x^2/2! + 20*x^3/3! + 132*x^4/4! + 1144*x^5/5! + 12600*x^6/6! +...
B(5,x) = x + 5*x^2/2! + 30*x^3/3! + 225*x^4/4! + 2120*x^5/5! + 24670*x^6/6! +...
B(6,x) = x + 6*x^2/2! + 42*x^3/3! + 354*x^4/4! + 3624*x^5/5! + 44700*x^6/6! +...
...
B(2,x)^2/2! = x^2/2! + 6*x^3/3! + 36*x^4/4! + 270*x^5/5! + 2604*x^6/6! +...
B(3,x)^3/3! = x^3/3! + 18*x^4/4! + 255*x^5/5! + 3600*x^6/6! + 54747*x^7/7! +...
B(4,x)^4/4! = x^4/4! + 40*x^5/5! + 1120*x^6/6! + 28140*x^7/7! + 693504*x^8/8! +...
B(5,x)^5/5! = x^5/5! + 75*x^6/6! + 3675*x^7/7! + 152250*x^8/8! + 5866245*x^9/9! +...
B(6,x)^6/6! = x^6/6! + 126*x^7/7! + 9912*x^8/8! + 634284*x^9/9! + 36483048*x^10/10! +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, intformal(A^m + x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A234855.

Sequence in context: A213236 A000169 A112319 * A038038 A368790 A048801

Adjacent sequences: A232549 A232550 A232551 * A232553 A232554 A232555

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2013

Status

approved