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A222014 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n^2) / Product_{k=1..n} (1 + k*x*A(x)^n). 1
1, 1, 2, 9, 54, 392, 3264, 30375, 311482, 3492134, 42613740, 564395954, 8094807168, 125423821396, 2093539627292, 37521869868373, 719483654260090, 14705046942685816, 319171681858506880, 7331367124418082012, 177646903957002411656, 4527740283395695051578 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the g.f. to the identities:

(1) 1/(1-x) = Sum_{n>=0} n!*x^n / Product_{k=1..n} (1 + k*x).

(2) C(x) = Sum_{n>=0} n!*x^n*C(x)^n / Product_{k=1..n} (1 + k*x*C(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

LINKS

Table of n, a(n) for n=0..21.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 54*x^4 + 392*x^5 + 3264*x^6 +...

where

A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)^4/((1+x*A(x)^2)*(1+2*x*A(x)^2)) + 3!*x^3*A(x)^9/((1+x*A(x)^3)*(1+2*x*A(x)^3)*(1+3*x*A(x)^3))  + 4!*x^4*A(x)^16/((1+x*A(x)^4)*(1+2*x*A(x)^4)*(1+3*x*A(x)^4)*(1+4*x*A(x)^4)) +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(m^2)/prod(k=1, m, 1+k*x*(A+x*O(x^n))^m))); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A222013, A221585.

Sequence in context: A089436 A000168 A307442 * A321974 A127128 A254795

Adjacent sequences:  A222011 A222012 A222013 * A222015 A222016 A222017

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2013

STATUS

approved

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Last modified July 10 11:42 EDT 2020. Contains 335576 sequences. (Running on oeis4.)