A357227 - OEIS

%I #22 Jun 07 2023 03:38:12

%S 1,1,5,27,156,961,6145,40546,273784,1883468,13153544,93012247,

%T 664640794,4791939802,34816034143,254659426691,1873698891024,

%U 13858201221637,102975937795619,768385165594607,5755185884844403,43253819566052165,326093530416255178,2465456045342545908

%N a(n) = coefficient of x^n, n >= 0, in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

%H Paul D. Hanna, <a href="/A357227/b357227.txt">Table of n, a(n) for n = 0..400</a>

%F Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

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

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

%e G.f.: A(x) = 1 + x + 5*x^2 + 27*x^3 + 156*x^4 + 961*x^5 + 6145*x^6 + 40546*x^7 + 273784*x^8 + 1883468*x^9 + 13153544*x^10 + 93012247*x^11 + 664640794*x^12 + ...

%e where

%e 1 = ... + x^(-3)/(2*A(x) - x^(-3))^4 + x^(-2)/(2*A(x) - x^(-2))^3 + x^(-1)/(2*A(x) - x^(-1))^2 + 1/(2*A(x) - 1) + x + x^2*(2*A(x) - x^2) + x^3*(2*A(x) - x^3)^2 + x^4*(2*A(x) - x^4)^3 + ... + x^n*(2*A(x) - x^n)^(n-1) + ...

%e SPECIFIC VALUES.

%e A(1/9) = 1.30108724398914093656591796643458817060949...

%e A(1/10) = 1.22176622612326449515553495048940456186175...

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

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

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

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

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

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

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

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

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

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

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

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

%Y Cf. A358961, A358962, A358963, A358964, A358965, A358937.

%Y Cf. A355868, A357232, A355865.

%Y Cf. A363312, A363313, A363314, A363315.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 17 2022