This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213591 #41 Jun 05 2025 09:55:42
%S A213591 1,1,4,24,178,1512,14152,142705,1528212,17211564,202460400,2474708496,
%T A213591 31310415376,408815254832,5495451727376,75907303147652,
%U A213591 1075685334980240,15618612118252960,232102241507321384,3526880759915999016,54755450619399484512,867928449982022915984
%N A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.
%C A213591 Unsigned version of A139702.
%C A213591 Self-convolution is A276370.
%C A213591 Row sums of triangle A277295.
%H A213591 Vaclav Kotesovec, <a href="/A213591/b213591.txt">Table of n, a(n) for n = 1..300</a>
%F A213591 G.f. satisfies:
%F A213591 (1) A(x) = x + A(A(x))^2.
%F A213591 (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
%F A213591 (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
%F A213591 (4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
%F A213591 (5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
%F A213591 (6) x = A(A( x-x^2 - A(x)^2 )). - _Paul D. Hanna_, Jul 01 2012
%F A213591 (7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
%F A213591 A = x + B^2;
%F A213591 B = A + C^2;
%F A213591 C = B + D^2;
%F A213591 D = C + E^2; ...
%F A213591 where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
%F A213591 ...
%F A213591 a(n) = Sum_{k=0..n-1} A277295(n,k).
%F A213591 From _Seiichi Manyama_, Jun 05 2025: (Start)
%F A213591 Let b(n,k) = [x^n] (A(x)/x)^k.
%F A213591 b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,2*j).
%F A213591 a(n) = b(n-1,1). (End)
%e A213591 G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
%e A213591 where A(x) = x + A(A(x))^2:
%e A213591 A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
%e A213591 A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
%e A213591 The g.f. satisfies the series:
%e A213591 A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
%e A213591 Logarithmic series:
%e A213591 log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
%e A213591 Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
%e A213591 G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
%e A213591 Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
%e A213591 F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...
%t A213591 terms = 22; A[_] = 0; Do[A[x_] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* _Jean-François Alcover_, Jan 09 2018 *)
%o A213591 (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
%o A213591 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A213591 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
%o A213591 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A213591 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
%o A213591 for(n=1,21,print1(a(n),", "))
%o A213591 (PARI) b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j)));
%o A213591 a(n) = b(n-1, 1); \\ _Seiichi Manyama_, Jun 05 2025
%Y A213591 Cf. A220379, A139702, A212411, A138740, A213628, A213639.
%Y A213591 Cf. A088714, A088717, A091713, A120971, A140094, A140095.
%Y A213591 Cf. A143426, A087949, A143435, A182969.
%Y A213591 Cf. A275765, A276360, A276361, A276362, A276363, A276370.
%Y A213591 Cf. A277295 (triangle).
%K A213591 nonn
%O A213591 1,3
%A A213591 _Paul D. Hanna_, Jun 14 2012