A303057 - OEIS

%I #13 Oct 06 2020 05:45:37

%S 1,1,3,21,221,3117,54597,1136127,27293715,742143113,22512196673,

%T 753402861159,27571631761077,1095346704175755,46948527167219957,

%U 2159638211148320085,106129271000784614099,5549226963359699829711,307623817602110038648839,18022345501064909362595723,1112657716434830018636702797

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

%H Paul D. Hanna, <a href="/A303057/b303057.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^(n+1).

%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1). - _Paul D. Hanna_, Dec 13 2018

%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.212154215724410476311... - _Vaclav Kotesovec_, Oct 06 2020

%e G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 221*x^4 + 3117*x^5 + 54597*x^6 + 1136127*x^7 + 27293715*x^8 + 742143113*x^9 + 22512196673*x^10 + ...

%e such that

%e A(x) = 1 + ((1+x)-1)/A(x) + ((1+x)^2-1)^2/A(x)^2 + ((1+x)^3-1)^3/A(x)^3 + ((1+x)^4-1)^4/A(x)^4 + ((1+x)^5-1)^5/A(x)^5 + ((1+x)^6-1)^6/A(x)^6 + ...

%e also,

%e 1 = 1/(A(x) + 1) + (1+x)/(A(x) + (1+x))^2 + (1+x)^4/(A(x) + (1+x)^2)^3 + (1+x)^9/(A(x) + (1+x)^3)^4 + (1+x)^16/(A(x) + (1+x)^4)^5 + (1+x)^25/(A(x) + (1+x)^5)^6 + (1+x)^36/(A(x) + (1+x)^6)^7 + ... + (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1) + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = Vec(sum(n=0,#A, ((1+x)^n - 1 +x*O(x^#A))^n / Ser(A)^(n+1) ) )[#A] );A[n+1]}

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

%Y Cf. A303058.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 20 2018