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.
1, 1, 6, 66, 981, 17576, 359101, 8109026, 198480901, 5197916551, 144326504376, 4220683214771, 129349023338616, 4138098976882836, 137770738524681831, 4761510500867829696, 170476220596490911691, 6311806909067054474161, 241301669599996597349506, 9512867537981387958217696, 386276838889195561353811321
Offset: 1
Keywords
Examples
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 + ...
where A(x) = x + x*A^6(x).
RELATED SERIES.
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 + ...
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 + ...
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 + ...
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 + ...
...
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 + ...
By formula (4),
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) + ...
Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+6)(x)):
n=0: x = A(x)/(1 + A(A(A(A(A(A(x))))))),
n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(A(x)))))))),
n=2: A(A(x)) = A(A(A(x)))/(1 + A(A(A(A(A(A(A(A(x))))))))),
n=3: A(A(A(x))) = A(A(A(A(x))))/(1 + A(A(A(A(A(A(A(A(A(x)))))))))),
...
Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)):
n=1: A(x) = x*(1 + A(A(A(A(A(A(x))))))),
n=2: A(A(x)) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
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))))))))),
...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..400
Programs
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PARI
/* By definition, A(x) = x + x*A(A(A(A(A(A(x)))))) */ /* Define the n-th iteration of function F: */ {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G} {a(n) = my(A=x); for(i=1, n, A = x + x*ITERATE(6, A, n)); polcoef(A, n)} for(n=1,30, print1(a(n),", "))
Formula
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.
(1) A(x) = x + x*A^6(x).
(2) A(x) = A(A(x))/(1 + A^7(x)).
(3) A(x) = Series_Reversion( x/(1 + A^5(x)) ).
(4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(5*k)(x).
(5) A^n(x) = A^(n+1)(x) / (1 + A^(n+6)(x)) for n >= 0.
(6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)) for n >= 1.
Comments