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A358963
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a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).
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6
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1, 2, 7, 31, 143, 731, 3896, 21444, 120967, 695699, 4063879, 24045306, 143808836, 867972228, 5280039896, 32339575813, 199266229047, 1234340158837, 7682216027973, 48014943810066, 301247658649431, 1896587278353158, 11978138505184044, 75867527248248561
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OFFSET
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0,2
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COMMENTS
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Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(4*n+3))^n, which holds formally for all y.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).
(2) x^3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n^2) / (1 - x^(4*n-3)*A(x))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} x^(5*n+3)* (A(x) - x^(4*n+3))^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n*(n-1)) / (1 - x^(4*n-3)*A(x))^(n+1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n*(n-1)) / (1 - x^(4*n-3)*A(x))^n.
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 7*x^2 + 31*x^3 + 143*x^4 + 731*x^5 + 3896*x^6 + 21444*x^7 + 120967*x^8 + 695699*x^9 + 4063879*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-5))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^3)^(-1) + x*(A - x^7)^0 + x^2*(A - x^11) + x^3*(A - x^15)^2 + x^4*(A - x^19)^3 + ... + x^n * (A - x^(4*n+3))^(n-1) + ...
also,
A(x) = ... + x^48/(1 - x^(-15)*A)^(-2) - x^24/(1 - x^(-11)*A)^(-1) + x^8 - 1/(1 - x^(-3)*A) + 1/(1 - x*A)^2 - x^8/(1 - x^5*A)^3 + x^24/(1 - x^9*A)^4 - x^48/(1 - x^13*A)^5 + ... + (-1)^(n+1)*x^(4*n*(n-1))/(1 - x^(4*n-3)*A)^(n+1) + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (Ser(A) - x^(4*n+3))^(n-1) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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