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

Author

Paul D. Hanna, Nov 26 2013

Keywords

• nonn

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).

Examples

			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! +...
		

Links

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

Crossrefs

Cf. A234855.

Programs

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),", "))

Formula

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