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 A378576 #10 Dec 03 2024 08:57:00
%S A378576 1,1,6,66,981,17576,359101,8109026,198480901,5197916551,144326504376,
%T A378576 4220683214771,129349023338616,4138098976882836,137770738524681831,
%U A378576 4761510500867829696,170476220596490911691,6311806909067054474161,241301669599996597349506,9512867537981387958217696,386276838889195561353811321
%N A378576 G.f. satisfies A(x) = x + x*A(A(A(A(A(A(x)))))), so that this sequence shifts left under the 6th self-COMPOSE.
%C A378576 Conjecture: a(n) == 1 (mod 5) for n >= 1.
%H A378576 Paul D. Hanna, <a href="/A378576/b378576.txt">Table of n, a(n) for n = 1..400</a>
%F A378576 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, wherein A^n(x) denotes the n-th iteration of A(x) with A^0(x) = x.
%F A378576 (1) A(x) = x + x*A^6(x).
%F A378576 (2) A(x) = A(A(x))/(1 + A^7(x)).
%F A378576 (3) A(x) = Series_Reversion( x/(1 + A^5(x)) ).
%F A378576 (4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(5*k)(x).
%F A378576 (5) A^n(x) = A^(n+1)(x) / (1 + A^(n+6)(x)) for n >= 0.
%F A378576 (6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)) for n >= 1.
%e A378576 G.f.: A(x) = x + x^2 + 6*x^3 + 66*x^4 + 981*x^5 + 17576*x^6 + 359101*x^7 + 8109026*x^8 + 198480901*x^9 + 5197916551*x^10 + ...
%e A378576 where A(x) = x + x*A^6(x).
%e A378576 RELATED SERIES.
%e A378576 A^2(x) = A(A(x)) = x + 2*x^2 + 14*x^3 + 163*x^4 + 2496*x^5 + 45577*x^6 + 944034*x^7 + 21546983*x^8 + 532083781*x^9 + 14041137012*x^10 + ...
%e A378576 A^3(x) = A(A(A(x))) = x + 3*x^2 + 24*x^3 + 297*x^4 + 4711*x^5 + 88073*x^6 + 1856179*x^7 + 42949757*x^8 + 1072699696*x^9 + 28584771073*x^10 + ...
%e A378576 A^4(x) = A(A(A(A(x)))) = x + 4*x^2 + 36*x^3 + 474*x^4 + 7816*x^5 + 150144*x^6 + 3230016*x^7 + 75976854*x^8 + 1923771726*x^9 + 51873235544*x^10 + ...
%e A378576 A^5(x) = A(A(A(A(A(x))))) = x + 5*x^2 + 50*x^3 + 700*x^4 + 12025*x^5 + 238000*x^6 + 5240145*x^7 + 125605650*x^8 + 3231333675*x^9 + 88338524825*x^10 + ...
%e A378576 ...
%e A378576 A^6(x) = A(A(A(A(A(A(x)))))) = x + 6*x^2 + 66*x^3 + 981*x^4 + 17576*x^5 + 359101*x^6 + 8109026*x^7 + 198480901*x^8 + 5197916551*x^9 + 144326504376*x^10 + ...
%e A378576 By formula (4),
%e A378576 A(x) = x + x*A^5(x) + x*A^5(x)*A^10(x) + x*A^5(x)*A^10(x)*A^15(x) + x*A^5(x)*A^10(x)*A^15(x)*A^20(x) + ...
%e A378576 Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+6)(x)):
%e A378576 n=0: x = A(x)/(1 + A(A(A(A(A(A(x))))))),
%e A378576 n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(A(x)))))))),
%e A378576 n=2: A(A(x)) = A(A(A(x)))/(1 + A(A(A(A(A(A(A(A(x))))))))),
%e A378576 n=3: A(A(A(x))) = A(A(A(A(x))))/(1 + A(A(A(A(A(A(A(A(A(x)))))))))),
%e A378576 ...
%e A378576 Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)):
%e A378576 n=1: A(x) = x*(1 + A(A(A(A(A(A(x))))))),
%e A378576 n=2: A(A(x)) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
%e A378576 n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x))))))))*(1 + A(A(A(A(A(A(A(A(x))))))))),
%e A378576 ...
%o A378576 (PARI) /* By definition, A(x) = x + x*A(A(A(A(A(A(x)))))) */
%o A378576 /* Define the n-th iteration of function F: */
%o A378576 {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
%o A378576 {a(n) = my(A=x); for(i=1, n, A = x + x*ITERATE(6, A, n)); polcoef(A, n)}
%o A378576 for(n=1,30, print1(a(n),", "))
%Y A378576 Cf. A030266, A091713, A196523, A378575.
%K A378576 nonn
%O A378576 1,3
%A A378576 _Paul D. Hanna_, Dec 01 2024