A303927 -id:A303927 - cp's OEIS frontend

A303926 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.

1, 1, 2, 12, 130, 1912, 34715, 743217, 18255118, 505070221, 15532353184, 525533183871, 19403298048040, 776437898905606, 33479679336072541, 1547841068340501230, 76390272348430998076, 4008960603544297652028, 222949077434693015546579, 13098226217965693342007714, 810657425687536689904281842

Offset: 0

Paul D. Hanna, May 03 2018

nonn

Compare to: 1 = Sum_{n>=0} ( (1 + x*G(x))^k - G(x) )^n holds trivially for fixed k>0 when G(x) = (1 + x*G(x))^k ; this sequence explores the case when k varies with n.

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 130*x^4 + 1912*x^5 + 34715*x^6 + 743217*x^7 + 18255118*x^8 + 505070221*x^9 + 15532353184*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)) - A(x)) + ((1 + x*A(x))^2 - A(x))^2 + ((1 + x*A(x))^3 - A(x))^3 + ((1 + x*A(x))^4 - A(x))^4 + ((1 + x*A(x))^5 - A(x))^5 + ...

Cf. A303927, A303928, A303923, A303056.

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

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.
(2) 1 = Sum_{n>=0} (1 + x*A(x))^(n^2) / (1 + A(x)*(1 + x*A(x))^n)^(n+1). - Paul D. Hanna, Dec 06 2018
G.f.: 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( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 03 2018

nonn

			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 + ...

Cf. A303926, A303927.

{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),", "))

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.

cp's OEIS Frontend

A303926 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.

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