A378829 - OEIS

%I #11 Dec 14 2024 07:16:38

%S 1,-2,5,-13,30,-74,202,-616,2126,-7828,29366,-110398,414214,-1556848,

%T 5892713,-22524354,86954484,-338421674,1324660464,-5204326208,

%U 20498580511,-80907096678,320002290542,-1268500509496,5040195484362,-20073242195580,80120884387322,-320442284717582,1283939790460139

%N G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

%C A signed version of A359673.

%H Paul D. Hanna, <a href="/A378829/b378829.txt">Table of n, a(n) for n = 1..303</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

%F (2) 1 = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2*x)^(n-1).

%F (3) 0 = Sum_{n=-oo..+oo} (-1)^n * (A(x)^n - 2*x)^(n-1).

%F (4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(2*n) * (A(x)^n + 2*x)^(n+1).

%F (5) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*x*A(x)^n)^n.

%F (6) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*x*A(x)^n)^(n+1).

%F (7) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n*(n+1)) / (1 + 2*x*A(x)^n)^(n+1).

%e G.f.: A(x) = x - 2*x^2 + 5*x^3 - 13*x^4 + 30*x^5 - 74*x^6 + 202*x^7 - 616*x^8 + 2126*x^9 - 7828*x^10 + 29366*x^11 - 110398*x^12 + ...

%e where 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

%e SPECIFIC VALUES.

%e A(t) = 1/6 at t = 0.24134833288352420167420358490093379236139061653959...

%e   where 1 = Sum_{n=-oo..+oo} (1/6^n - 2*t)^n.

%e A(t) = 1/7 at t = 0.19473287649699543474178954182484954936895675300220...

%e   where 1 = Sum_{n=-oo..+oo} (1/7^n - 2*t)^n.

%e A(t) = 1/8 at t = 0.16330047299490635761734791354706359079698287572429...

%e   where 1 = Sum_{n=-oo..+oo} (1/8^n - 2*t)^n.

%e A(t) = exp(-Pi) at t = 0.04720243920412572796492634515550526365563452970121157309...

%e   where 1 = Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*t)^n,

%e   also, 1 = Sum_{n=-oo..+oo} exp(-n^2*Pi) / (1 - 2*t*exp(-n*Pi))^n;

%e   compare to Sum_{n=-oo..+oo} exp(-n^2*Pi) = Pi^(1/4)/gamma(3/4).

%e A(t) = exp(-2*Pi) at t = 0.001874436990256710694689538031391789940066981740061145959...

%e   where 1 = Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*t)^n,

%e   also, 1 = Sum_{n=-oo..+oo} exp(-2*n^2*Pi) / (1 - 2*t*exp(-2*n*Pi))^n;

%e   compare to Sum_{n=-oo..+oo} exp(-2*n^2*Pi) = Pi^(1/4)/gamma(3/4) * sqrt(2+sqrt(2))/2.

%e A(1/5) = 0.14570268760195709902234365534810153966906514204980...

%e   where 1 = Sum_{n=-oo..+oo} (A(1/5)^n - 2/5)^n.

%e A(1/6) = 0.12698642862956730423090954809810167590805619510041...

%e   where 1 = Sum_{n=-oo..+oo} (A(1/6)^n - 1/3)^n.

%e A(1/7) = 0.11253270334433369822784652362071431711460474251926...

%e A(1/8) = 0.10104551587569245791494155789285565556961920656039...

%e   where 1 = Sum_{n=-oo..+oo} (A(1/8)^n - 1/4)^n.

%o (PARI) {a(n) = my(V=[0,1],A); for(i=1, n, V=concat(V, 0); A=Ser(V);

%o V[#V] = polcoef( -sum(m=-#V, #V, (A^m - 2*x)^m ), #V-1)/2); V[n+1]}

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

%Y Cf. A355868, A359673.

%K sign

%O 1,2

%A _Paul D. Hanna_, Dec 13 2024