A303928 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.

1, 1, 4, 29, 312, 4454, 78649, 1644280, 39580036, 1076460972, 32628557331, 1090654903233, 39861817143230, 1581648436369772, 67718096677762406, 3112120229328860775, 152815413664021339930, 7985028281346030147672, 442406826626726978612624, 25906474516335623637923581, 1598761621228278791567817906

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( (1 + x*A(x)^3)^n - A(x) )^n.

(2) 1 = Sum_{n>=0} (1 + x*A(x)^3)^(n^2) / (1 + A(x)*(1 + x*A(x)^3)^n)^(n+1). - Paul D. Hanna, Dec 11 2018

G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x)^2)^n - F(x))^n, where F(x) is the g.f. of A303927.

G.f.: sqrt( 1/x*Series_Reversion( x/G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^n - G(x))^n, where G(x) is the g.f. of A303926.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 312*x^4 + 4454*x^5 + 78649*x^6 + 1644280*x^7 + 39580036*x^8 + 1076460972*x^9 + 32628557331*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)^3) - A(x)) + ((1 + x*A(x)^3)^2 - A(x))^2 + ((1 + x*A(x)^3)^3 - A(x))^3 + ((1 + x*A(x)^3)^4 - A(x))^4 + ((1 + x*A(x)^3)^5 - A(x))^5 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( (1 + x*Ser(A)^3)^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303926, A303927.

Sequence in context: A210949 A014622 A030019 * A201627 A195194 A215955

Adjacent sequences: A303925 A303926 A303927 * A303929 A303930 A303931

Keyword

nonn

Author

Paul D. Hanna, May 03 2018

Status

approved