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 A365516 #27 Oct 07 2023 22:23:32 %S A365516 1,2,4,10,32,112,400,1464,5520,21296,83456,331136,1328320,5379200, %T A365516 21959936,90271904,373347840,1552438016,6486311680,27217331456, %U A365516 114649525760,484640538112,2055185596416,8740711936000,37273693649920,159340373710848,682708771254272,2931290431277056 %N A365516 Expansion of g.f. A(x) satisfying A(x) = (1 + 2*x*A(x))^2 / (1 + 2*x*A(x) - 2*x^3*A(x)^3). %C A365516 Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108): %C A365516 (1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1. %C A365516 (2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1. %C A365516 Related identity: [x^(n-1)] (1 + (n+1)*x*B(x))^n / B(x)^n = n*(n-1)^(n-1) for n >= 1 when B(x) = 1/(1 - 2*x). %C A365516 Related identity: F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1), which holds formally for all Maclaurin series F(x). - _Paul D. Hanna_, Oct 03 2023 %H A365516 Paul D. Hanna, <a href="/A365516/b365516.txt">Table of n, a(n) for n = 0..400</a> %F A365516 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas. %F A365516 (1) A(x) = (1 + 2*x*A(x))^2 / (1 + 2*x*A(x) - 2*x^3*A(x)^3). %F A365516 (2) A(x/B(x)) = B(x) where B(x) = (1 + 2*x)^2 / (1 + 2*x - 2*x^3). %F A365516 (3) A(x) = (1/x) * Series_Reversion( x*(1 + 2*x - 2*x^3)/(1 + 2*x)^2 ). %F A365516 (4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+1)^(n-2) for n > 1. %F A365516 (5) [x^(n-1)] (1 + (n-1)*x*A(x))^n / A(x)^n = -n*(n-3)^(n-2) for n > 1. %F A365516 (6) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = (n^(n-1) - n*(n-2)^(n-2))/2 for n >= 1. %F A365516 (7) A(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1). - _Paul D. Hanna_, Oct 03 2023 %e A365516 G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 32*x^4 + 112*x^5 + 400*x^6 + 1464*x^7 + 5520*x^8 + 21296*x^9 + 83456*x^10 + ... %e A365516 where A(x) = (1 + 2*x*A(x))^2/(1 + 2*x*A(x) - 2*x^3*A(x)^3). %e A365516 RELATED SERIES. %e A365516 B(x) = A(x/B(x)) where B(x) = (1 + 2*x)^2/(1 + 2*x - 2*x^3) begins %e A365516 B(x) = 1 + 2*x + 2*x^3 + 4*x^6 - 8*x^7 + 16*x^8 - 24*x^9 + 32*x^10 - 32*x^11 + 16*x^12 + 32*x^13 - 128*x^14 + ... %e A365516 RELATED TABLE. %e A365516 The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins: %e A365516 n=1: [1, 0, 0, -2, -8, -24, -76, -272, ...]; %e A365516 n=2: [1, 2, 1, -4, -20, -64, -196, -664, ...]; %e A365516 n=3: [1, 6, 12, 2, -48, -192, -600, -1896, ...]; %e A365516 n=4: [1, 12, 54, 100, -23, -600, -2224, -6944, ...]; %e A365516 n=5: [1, 20, 160, 630, 1080, -696, -8660, -30960, ...]; %e A365516 n=6: [1, 30, 375, 2488, 9027, 14406, -15371, -144852, ...]; %e A365516 n=7: [1, 42, 756, 7546, 44800, 153552, 229376, -342728, ...]; %e A365516 n=8: [1, 56, 1372, 19192, 167222, 921400, 3026332, 4251528, ...]; ... %e A365516 in which the main diagonal equals n*(n+1)^(n-2) for n > 1. %t A365516 a[n_] := Module[{A = {1}, m, x}, Do[AppendTo[A, 0]; m = Length[A]; A[[-1]] = Coefficient[(1 + (m + 1)*x*SeriesData[x, 0, A, 0, m + 1, 1])^m/SeriesData[x, 0, A, 0, m + 1, 1]^m, x, m - 1]/m - (m + 1)^(m - 2), {i, 1, n}]; A[[n + 1]]]; Table[a[n], {n, 0, 27}] (* _Robert P. P. McKone_, Sep 07 2023 *) %o A365516 (PARI) /* Using series reversion */ %o A365516 {a(n) = my(A = (1/x)*serreverse(x*(1 + 2*x - 2*x^3)/(1 + 2*x +x*O(x^n))^2)); %o A365516 polcoeff(A,n)} %o A365516 for(n=0,30,print1(a(n),", ")) %o A365516 (PARI) /* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n/A(x)^n = n*(n+1)^(n-2) */ %o A365516 {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; %o A365516 A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+1)^(m-2) );A[n+1]} %o A365516 for(n=0,30,print1(a(n),", ")) %Y A365516 Cf. A365574, A303063, A365095. %K A365516 nonn %O A365516 0,2 %A A365516 _Paul D. Hanna_, Sep 07 2023