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 A383563 #18 Jun 01 2025 19:39:39
%S A383563 1,1,1,3,13,72,465,3362,26531,224856,2024188,19202830,190857879,
%T A383563 1978567663,21319434418,238109360460,2750229390071,32789591062124,
%U A383563 402891169846242,5094855923807780,66229610059651788,884081025776797026,12107164229698851942,169954380180177899277,2443554376412586234247
%N A383563 G.f. A(x) satisfies A( x*(1+x)/A(x)^2 ) = 1 + x.
%H A383563 Paul D. Hanna, <a href="/A383563/b383563.txt">Table of n, a(n) for n = 0..470</a>
%F A383563 G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145345 and C(x) = g.f. of A121687 satisfies the following formulas.
%F A383563 (1) A( x*(1+x)/A(x)^2 ) = 1 + x.
%F A383563 (2.a) Series_Reversion( x/A(x) ) = x + x*Series_Reversion( x/A(x)^2 ).
%F A383563 (2.b) [x^n] A(x)^(n+1)/(n+1) = [x^(n-1)] A(x)^(2*n)/n for n >= 1.
%F A383563 (2.c) B(x) = 1 + x*C(x)^2.
%F A383563 (3.a) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C(x/B(x)).
%F A383563 (3.b) A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) = B(x*C(x)).
%F A383563 (4.a) A(x) = A(x)^2 - x*C(x/A(x))^2.
%F A383563 (4.b) B( x/A(x)^2 ) = 1 + x.
%F A383563 (4.c) C( (x/(1+x))/A(x)^2 ) = 1 + x.
%F A383563 (4.d) B( (x/(1+x))/A(x)^2 ) = 1 + x*(1+x)/A(x)^2.
%F A383563 (4.e) A( (x/(1+x))/A(x)^2 + x^2/A(x)^4 ) = 1 + x*(1+x)/A(x)^2.
%F A383563 (5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
%F A383563 (5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
%F A383563 (5.c) A145345(n) = [x^(n-1)] C(x)^2 = [x^(n-1)] A(x)^(2*n)/n for n >= 1.
%e A383563 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 72*x^5 + 465*x^6 + 3362*x^7 + 26531*x^8 + 224856*x^9 + 2024188*x^10 + ...
%e A383563 where A( x*(1+x)/A(x)^2 ) = 1 + x.
%e A383563 RELATED SERIES.
%e A383563 The g.f. of A145345 begins
%e A383563 B(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + 10706*x^7 + 89120*x^8 + 794347*x^9 + ...
%e A383563 where B(x/B(x)) = 1 + x*B(x)
%e A383563 also, B( x/A(x)^2 ) = 1 + x.
%e A383563 The g.f. of A121687 begins
%e A383563 C(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + 37244*x^7 + 335153*x^8 + 3194510*x^9 + ...
%e A383563 where C(x) = 1/(1 - x*C(x*C(x))^2)
%e A383563 also, C( (x/(1+x))/A(x)^2 ) = 1 + x.
%e A383563 C(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 + 89120*x^7 + 794347*x^8 + ...
%e A383563 where B(x) = 1 + x*C(x)^2.
%o A383563 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A383563 A[#A] = polcoef(x - serreverse(x/Ser(A)) + x*serreverse(x/Ser(A)^2),#A) ); A[n+1]}
%o A383563 for(n=0,30,print1(a(n),", "))
%Y A383563 Cf. A384265, A145345 (B(x)), A121687 (C(x)).
%K A383563 nonn
%O A383563 0,4
%A A383563 _Paul D. Hanna_, May 26 2025