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 A376176 #13 Jun 05 2025 09:55:34
%S A376176 1,1,6,55,622,8015,113164,1711898,27357970,457507917,7952476482,
%T A376176 142972019125,2648639456048,50415218306637,983728646223556,
%U A376176 19641163430509505,400671660024507294,8340743906266061866,176998642509849677206,3825680705425292568049,84159282700462688412042
%N A376176 G.f. A(x) satisfies x = A( x - A(x)^4/x^2 ).
%H A376176 Paul D. Hanna, <a href="/A376176/b376176.txt">Table of n, a(n) for n = 1..300</a>
%F A376176 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A376176 (1) x = A( x - A(x)^4/x^2 ).
%F A376176 (2) A(x)^3 = x*A(x)^2 + A(A(x))^4.
%F A376176 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n) / n!.
%F A376176 (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n+1) / n! ).
%F A376176 From _Seiichi Manyama_, Jun 05 2025: (Start)
%F A376176 Let b(n,k) = [x^n] (A(x)/x)^k.
%F A376176 b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,4*j).
%F A376176 a(n) = b(n-1,1). (End)
%e A376176 G.f.: A(x) = x + x^2 + 6*x^3 + 55*x^4 + 622*x^5 + 8015*x^6 + 113164*x^7 + 1711898*x^8 + 27357970*x^9 + 457507917*x^10 + ...
%e A376176 where x = A( x - A(x)^4/x^2 ).
%e A376176 RELATED SERIES.
%e A376176 A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 122*x^5 + 1390*x^6 + 17934*x^7 + 252847*x^8 + 3814724*x^9 + ...
%e A376176 A(x)^3 = x^3 + 3*x^4 + 21*x^5 + 202*x^6 + 2322*x^7 + 30030*x^8 + 423111*x^9 + 6369930*x^10 + ...
%e A376176 where A(x)^3 = x*A(x)^2 + A(A(x))^4.
%e A376176 A(x)^4 = x^4 + 4*x^5 + 30*x^6 + 296*x^7 + 3437*x^8 + 44600*x^9 + 628454*x^10 + 9446280*x^11 + ...
%e A376176 A(A(x))^4 = x^4 + 8*x^5 + 80*x^6 + 932*x^7 + 12096*x^8 + 170264*x^9 + 2555206*x^10 + 40413484*x^11 + ...
%e A376176 where A(x) = x + A(A(x))^4 / A(x)^2.
%e A376176 A(A(x)) = x + 2*x^2 + 14*x^3 + 141*x^4 + 1712*x^5 + 23392*x^6 + 347444*x^7 + 5498681*x^8 + 91552406*x^9 + ...
%e A376176 A(A(x))^2/A(x) = x + 3*x^2 + 23*x^3 + 242*x^4 + 3017*x^5 + 41965*x^6 + 631381*x^7 + 10089533*x^8 + 169256922*x^9 + ...
%o A376176 (PARI) {a(n) = my(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^4/x^2 +x*O(x^n))); polcoeff(A, n))}
%o A376176 for(n=1, 25, print1(a(n), ", "))
%o A376176 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A376176 {a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
%o A376176 for(n=1, 25, print1(a(n), ", "))
%o A376176 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A376176 {a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
%o A376176 for(n=1, 25, print1(a(n), ", "))
%o A376176 (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, 4*j)));
%o A376176 a(n) = b(n-1, 1); \\ _Seiichi Manyama_, Jun 05 2025
%Y A376176 Cf. A213591, A213639.
%K A376176 nonn
%O A376176 1,3
%A A376176 _Paul D. Hanna_, Sep 21 2024