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 A357161 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n. 7
 1, 1, 3, 15, 71, 378, 2087, 12006, 70910, 428021, 2627731, 16358961, 103027423, 655236314, 4202210514, 27145925685, 176474644608, 1153679423108, 7579526316199, 50017854059557, 331390828183765, 2203548061830875, 14700363755114949, 98363233394747546 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to A357151. Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1). Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..400 FORMULA G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations. (1) A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n. (2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ). (3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n. (4) -A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n. (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n. (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n. EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 71*x^4 + 378*x^5 + 2087*x^6 + 12006*x^7 + 70910*x^8 + 428021*x^9 + 2627731*x^10 + ... such that A(x) = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ... also -A(x)^4 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ... PROG (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = polcoeff(Ser(A) - sum(n=-#A\3-2, #A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n ), #A-2); ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A357151, A357160, A357162, A357163, A357164, A357165. Sequence in context: A178345 A183547 A123942 * A290902 A155117 A137638 Adjacent sequences: A357158 A357159 A357160 * A357162 A357163 A357164 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 17 2022 STATUS approved

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Last modified September 16 06:41 EDT 2024. Contains 375959 sequences. (Running on oeis4.)