cp's OEIS Frontend

A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.

%I A359721 #14 Mar 14 2023 04:44:06
%S A359721 1,1,3,10,37,127,460,1710,6461,24851,96921,382358,1522997,6116518,
%T A359721 24740564,100698617,412126133,1694982357,7001729420,29037602898,
%U A359721 120856092153,504647152650,2113469775619,8875358529059,37364827472930,157668052571948,666735804080597,2825054673048981
%N A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
%C A359721 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%H A359721 Paul D. Hanna, <a href="/A359721/b359721.txt">Table of n, a(n) for n = 0..300</a>
%F A359721 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A359721 (1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
%F A359721 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + x^n)^n * A(x)^n).
%F A359721 (3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k), for n >= 0.
%F A359721 a(n) ~ c * d^n / n^(3/2), where d = 4.470597712126170109... and c = 1.18164918660560739... - _Vaclav Kotesovec_, Mar 14 2023
%e A359721 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 127*x^5 + 460*x^6 + 1710*x^7 + 6461*x^8 + 24851*x^9 + 96921*x^10 + 382358*x^11 + 1522997*x^12 + ...
%e A359721 SPECIFIC VALUES.
%e A359721 A(2/9) = 2.24070435506724977359903344036738515875266644317987374...
%e A359721 A(x) = 2 at x = 0.21791735938682393028374635435485389216073583164032813...
%e A359721 A(1/5) = 1.63325728843716074555468074513852677972333543319428229...
%e A359721 A(1/6) = 1.36828627213340815002770404510072582545059876619425902...
%o A359721 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A359721 A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (1 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
%o A359721 for(n=0, 30, print1(a(n), ", "))
%Y A359721 Cf. A359720, A355357, A357797, A359723, A359724.
%K A359721 nonn
%O A359721 0,3
%A A359721 _Paul D. Hanna_, Jan 11 2023