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 A380554 #7 Jan 29 2025 12:46:08
%S A380554 1,1,1,1,2,6,16,36,75,163,391,991,2498,6150,15016,37116,93482,238154,
%T A380554 608074,1551370,3964200,10176384,26261500,68034484,176661828,
%U A380554 459534596,1197777556,3129475636,8195867902,21508247446,56540427826,148863643466,392539322259,1036662269875,2741706892035
%N A380554 G.f. A(x) satisfies A(x)^4 = A( A(x)^3 * x/(1-x) ).
%H A380554 Paul D. Hanna, <a href="/A380554/b380554.txt">Table of n, a(n) for n = 1..1030</a>
%F A380554 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380554 (1) A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ).
%F A380554 (2) A(x) = x * (1 + Sum_{n>=0} A(x)^(4^n)).
%F A380554 (3) A(x) = x/(1-x) * (1 + Sum_{n>=1} A(x)^(4^n)).
%F A380554 (4) A(x)^4 = A( A(x)^3 * x/(1-x) ).
%F A380554 (5) A(x)^16 = A( A(x)^15 * x/(1 - x - x*A(x)^3) ).
%F A380554 (6) A(x)^64 = A( A(x)^63 * x/(1 - x - x*A(x)^3 - x*A(x)^15) ).
%F A380554 (7) A(x)^(4^n) = A( A(x)^(4^n-1) * x/(1 - x*Sum_{k=0..n-1} A(x)^(4^k-1)) ) for n >= 1.
%F A380554 The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 4^n*A(r)^(4^n-1)) and A(r) = A( A(r)^3*r/(1-r) )^(1/4), where r = 0.3613437470225014946622689597447779556234350427479140... and A(r) = 0.7371720020640001613320630406857895231048184830453856...
%e A380554 G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 16*x^7 + 36*x^8 + 75*x^9 + 163*x^10 + 391*x^11 + 991*x^12 + 2498*x^13 + 6150*x^14 + 15016*x^15 + ...
%e A380554 where A(x)^4 = A( A(x)^3 * x/(1-x) );
%e A380554 also, A(x) = x*(1 + A(x) + A(x)^4 + A(x)^16 + A(x)^64 + ...).
%e A380554 RELATED SERIES.
%e A380554 x/(1 + x + x^4 + x^16 + x^64 + ...) = x - x^2 + x^3 - x^4 + x^6 - 2*x^7 + 3*x^8 - 3*x^9 + 2*x^10 - 3*x^12 + 6*x^13 - 8*x^14 + 8*x^15 - 5*x^16 + ...
%e A380554 where x = A( x/(1 + x + x^4 + x^16 + x^64 + ...) ).
%e A380554 A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 112*x^9 + 288*x^10 + ...
%e A380554 A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 39*x^8 + 88*x^9 + 228*x^10 + 600*x^11 + ...
%e A380554 SPECIFIC VALUES.
%e A380554 A(t) = 7/10 at t = 0.36018915820185609929548309671397017657231396...
%e A380554 where (7/10)^4 = A( (7/10)^3*t/(1-t) )
%e A380554 and t = (7/10)/(1 + Sum_{n>=0} (7/10)^(4^n)).
%e A380554 A(t) = 2/3 at t = 0.357324077294579321123715825007257976292387856...
%e A380554 where 16/81 = A( (8/27)*t/(1-t) )
%e A380554 and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(4^n)).
%e A380554 A(t) = 1/2 at t = 0.319996875030517280093584464262123092506355813...
%e A380554 where 1/16 = A( (1/8)*t/(1-t) )
%e A380554 and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(4^n)).
%e A380554 A(t) = 1/3 at t = 0.247706417742171319902767393551872977585317906...
%e A380554 where 1/81 = A( (1/27)*t/(1-t) )
%e A380554 and t = (1/3)/(1 + Sum_{n>=0} (1/3)^(4^n)).
%e A380554 A(1/3) = 0.54373202136840396341881074287828877295481851718413...
%e A380554 where A(1/3)^4 = A( A(1/3)^3*(1/2) ).
%e A380554 A(1/4) = 0.33766677567921691723942758840979376280294197783058...
%e A380554 where A(1/4)^4 = A( A(1/4)^3*(1/3) ).
%e A380554 A(1/5) = 0.25099215755350299738032710744403195608988446686839...
%e A380554 A(1/6) = 0.20032206620931060989695576481191496886558371212657...
%o A380554 (PARI) /* A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ) */
%o A380554 {a(n) = my(A = serreverse( x/(1 + sum(m=0,ceil(log(n+1)/log(4)), x^(4^m) +x*O(x^n))) )); polcoef(A,n)}
%o A380554 for(n=1, 40, print1(a(n), ", "))
%o A380554 (PARI) /* A(x)^4 = A( A(x)^3 * x/(1-x) ) */
%o A380554 {a(n) = my(A=[1], Ax);
%o A380554 for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
%o A380554 A[#A] = -polcoeff( Ax^4 - subst(Ax, x, Ax^3*x/(1-x) ), #A+3) ); A[n]}
%o A380554 for(n=1, 40, print1(a(n), ", "))
%Y A380554 Cf. A075864, A374565.
%K A380554 nonn
%O A380554 1,5
%A A380554 _Paul D. Hanna_, Jan 26 2025