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 A145347 #9 Jun 04 2025 17:20:45
%S A145347 1,1,4,26,220,2203,24836,306104,4047988,56713521,834286612,
%T A145347 12801754120,203889888832,3357619794321,56999146850380,
%U A145347 995081586539016,17830012791062632,327376145842252333,6151225530281186372,118142009771446643592,2317165307900630229384
%N A145347 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^3.
%C A145347 More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
%C A145347 A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
%C A145347 thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.
%H A145347 Paul D. Hanna, <a href="/A145347/b145347.txt">Table of n, a(n) for n = 0..300</a>
%F A145347 G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)).
%F A145347 a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(n+3)/(n+3) for n>=1 (see comment).
%e A145347 G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 +...
%e A145347 A(x)^3 = 1 + 3*x + 15*x^2 + 103*x^3 + 876*x^4 + 8679*x^5 + 96382*x^6 +...
%e A145347 A(x/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 876*x^5 + 8679*x^6 +...
%e A145347 A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)):
%e A145347 G(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 +...
%e A145347 To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3),
%e A145347 form a table of coefficients in A(x)^(n+3) as follows:
%e A145347 A^4: [(1), 4, 22, 156, 1337, 13220, 145988, 1759876, ...];
%e A145347 A^5: [1, (5), 30, 220, 1905, 18836, 207100, 2481740, ...];
%e A145347 A^6: [1, 6, (39), 296, 2595, 25704, 281727, 3358488, ...];
%e A145347 A^7: [1, 7, 49, (385), 3423, 34020, 372141, 4416658, ...];
%e A145347 A^8: [1, 8, 60, 488, (4406), 44000, 480900, 5686480, ...];
%e A145347 A^9: [1, 9, 72, 606, 5562, (55881), 610872, 7202268, ...]; ...
%e A145347 in which the main diagonal forms the initial terms of this sequence:
%e A145347 [4/4*(1), 4/5*(5), 4/6*(39), 4/7*(385), 4/8*(4406), 4/9*(55881), ...].
%o A145347 (PARI) {a(n) = my(F=1+x); for(i=0,n, G=serreverse(x/(F+x*O(x^n))^1)/x; F=1+x*G^4); polcoef(F,n)}
%o A145347 (PARI) /* This sequence is generated when k=1, m=3: A(x/A(x)^k) = 1 + x*A(x)^m */
%o A145347 {a(n, k=1, m=3) = my(A=sum(i=0, n-1, a(i, k, m)*x^i) +x*O(x^n)); if(n==0, 1, polcoef((m+k)/(m+k*n)*A^(m+k*n), n-1))}
%o A145347 for(n=0,20,print1(a(n),", "))
%Y A145347 Cf. A384265, A145348, A145350, A147664, A120972.
%K A145347 nonn
%O A145347 0,3
%A A145347 _Paul D. Hanna_, Nov 09 2008