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A357227
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).
16
1, 1, 5, 27, 156, 961, 6145, 40546, 273784, 1883468, 13153544, 93012247, 664640794, 4791939802, 34816034143, 254659426691, 1873698891024, 13858201221637, 102975937795619, 768385165594607, 5755185884844403, 43253819566052165, 326093530416255178, 2465456045342545908
OFFSET
0,3
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
LINKS
FORMULA
Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).
(2) 2*A(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - x^n)^(n-1).
(3) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*x^n*A(x))^(n+1).
(4) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*x^n*A(x))^(n+1).
EXAMPLE
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 + ...
where
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) + ...
SPECIFIC VALUES.
A(1/9) = 1.30108724398914093656591796643458817060949...
A(1/10) = 1.22176622612326449515553495048940456186175...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(m-1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 17 2022
STATUS
approved