The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A357160 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n. 7
 1, 1, 2, 8, 24, 88, 313, 1187, 4549, 17898, 71324, 288365, 1177729, 4856051, 20178061, 84427850, 355375253, 1503849591, 6394015744, 27301536104, 117020066991, 503313598572, 2171633107742, 9396938664272, 40769489510945, 177313714453588, 772906669281227, 3376119803594888 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to A356783. 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) 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n. (2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ). (3) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n. (4) -A(x)^3 = 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 + 2*x^2 + 8*x^3 + 24*x^4 + 88*x^5 + 313*x^6 + 1187*x^7 + 4549*x^8 + 17898*x^9 + 71324*x^10 + ... such that 1 = ... + 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)^3 = ... + 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(1 - 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. A356783, A357161, A357162, A357163, A357164, A357165. Sequence in context: A007223 A106189 A106183 * A362139 A060899 A213951 Adjacent sequences: A357157 A357158 A357159 * A357161 A357162 A357163 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 17 2022 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 9 18:46 EDT 2024. Contains 375765 sequences. (Running on oeis4.)