A352706 - OEIS

%I #9 Sep 01 2022 12:04:34

%S 1,3,13,65,351,1989,11650,69900,427167,2648438,16612947,105215448,

%T 671760933,4318468134,27926126553,181520036178,1185220461867,

%U 7769787812787,51117085998498,337373170647840,2233091755252871,14819626692452231,98582852467595847

%N G.f. A(x) satisfies: (1 - x*A(x))^7 = 1 - 7*x - x^7*A(x^7).

%C Essentially an unsigned version of A352705 (after dropping the initial term).

%H Paul D. Hanna, <a href="/A352706/b352706.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies:

%F (1) (1 + x*A(-x))^7 = 1 + 7*x + x^7*A(-x^7).

%F (2) A(x) = (1 - (1 - 7*x - x^7*A(x^7))^(1/7))/x.

%F (3) A(x)^7 = A(x^7) (mod 7).

%e G.f.: A(x) = 1 + 3*x + 13*x^2 + 65*x^3 + 351*x^4 + 1989*x^5 + 11650*x^6 + 69900*x^7 + 427167*x^8 + 2648438*x^9 + 16612947*x^10 + ...

%e where

%e (1 - x*A(x))^7 = 1 - 7*x - x^7 - 3*x^14 - 13*x^21 - 65*x^28 - 351*x^35 - 1989*x^42 - 11650*x^49 - 69900*x^56 - 427167*x^63 - 2648438*x^70 + ...

%e also

%e (1 - 7*x - x^7*A(x^7))^(1/7) = 1 - x - 3*x^2 - 13*x^3 - 65*x^4 - 351*x^5 - 1989*x^6 - 11650*x^7 - 69900*x^8 - 427167*x^9 - 2648438*x^10 + ...

%e which equals 1 - x*A(x).

%o (PARI) {a(n) = my(A=1+3*x); for(i=1,n,

%o A = (1 - (1 - 7*x - x^7*subst(A,x,x^7) + x*O(x^(n+1)))^(1/7))/x);

%o polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A352705, A352702, A352704.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 29 2022