A378575 - OEIS

%I #9 Dec 03 2024 08:57:04

%S 1,1,5,45,545,7945,132005,2423501,48224129,1026722489,23177970949,

%T 551133715197,13734995332769,357361170997321,9677345660994725,

%U 272075021315860781,7925076713952829697,238747406787319312025,7427421640015549840133,238301672444134819413533,7875799810817511976148129

%N G.f. satisfies A(x) = x + x*A(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE.

%C Conjecture: a(n) == 1 (mod 4) for n >= 1.

%H Paul D. Hanna, <a href="/A378575/b378575.txt">Table of n, a(n) for n = 1..400</a>

%F 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 (1) A(x) = x + x*A^5(x).

%F (2) A(x) = A(A(x))/(1 + A^6(x)).

%F (3) A(x) = Series_Reversion( x/(1 + A^4(x)) ).

%F (4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(4*k)(x).

%F (5) A^n(x) = A^(n+1)(x) / (1 + A^(n+5)(x)) for n >= 0.

%F (6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(x)) for n >= 1.

%e G.f.: A(x) = x + x^2 + 5*x^3 + 45*x^4 + 545*x^5 + 7945*x^6 + 132005*x^7 + 2423501*x^8 + 48224129*x^9 + 1026722489*x^10 + ...

%e where A(x) = x + x*A^5(x).

%e RELATED SERIES.

%e A^2(x) = A(A(x)) = x + 2*x^2 + 12*x^3 + 116*x^4 + 1460*x^5 + 21820*x^6 + 369140*x^7 + 6873732*x^8 + 138390908*x^9 + 2976373452*x^10 + ...

%e A^3(x) = A(A(A(x))) = x + 3*x^2 + 21*x^3 + 219*x^4 + 2885*x^5 + 44483*x^6 + 770269*x^7 + 14610939*x^8 + 298729077*x^9 + 6510526915*x^10 + ...

%e A^4(x) = A(A(A(A(x)))) = x + 4*x^2 + 32*x^3 + 360*x^4 + 4984*x^5 + 79648*x^6 + 1417768*x^7 + 27500512*x^8 + 572918728*x^9 + 12690763632*x^10 + ...

%e A^5(x) = A(A(A(A(A(x))))) = x + 5*x^2 + 45*x^3 + 545*x^4 + 7945*x^5 + 132005*x^6 + 2423501*x^7 + 48224129*x^8 + 1026722489*x^9 + ...

%e ...

%e By formula (4),

%e A(x) = x + x*A^4(x) + x*A^4(x)*A^8(x) + x*A^4(x)*A^8(x)*A^12(x) + x*A^4(x)*A^8(x)*A^12(x)*A^16(x) + ...

%e Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+5)(x)):

%e n=0: x = A(x)/(1 + A(A(A(A(A(x)))))),

%e n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(x))))))),

%e n=2: A(A(x)) = A(A(A(x)))/(1 + A(A(A(A(A(A(A(x)))))))),

%e n=3: A(A(A(x))) = A(A(A(A(x))))/(1 + A(A(A(A(A(A(A(A(x))))))))),

%e ...

%e Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(x)):

%e n=1: A(x) = x*(1 + A(A(A(A(A(x)))))),

%e n=2: A(A(x)) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x))))))),

%e n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),

%e ...

%o (PARI) /* By definition, A(x) = x + x*A(A(A(A(A(x))))) */

%o /* Define the n-th iteration of function F: */

%o {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}

%o {a(n) = my(A=x); for(i=1, n, A = x + x*ITERATE(5, A, n)); polcoef(A, n)}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A030266, A091713, A196523.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 01 2024