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 A357157 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2). 4
 1, 1, 1, 1, 7, 1, 1, 1, 22, 1, 1, -19, 57, 1, 1, 1, 22, 1, 1, 1, 303, -349, 1, 1, 463, 1, -593, 1, 793, 1, 1, -2204, 2584, 1, 1, 1, -2287, 1, 3082, 1, 3004, -8084, 1, 1, 14465, -3674, -14299, 1, 6189, 1, 22276, -24023, -2056, 1, 1, 1, 18714, 1, 1, -34985, 24305, -60059, 87517, 1, 20350 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,5 COMMENTS Related identities: (I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1). (I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1). (I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1). (I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1). (I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k. (I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1). (I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2). (I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3). (I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4). (I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k. (I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)*(n+2)*(n+3)/120 * x^(4*n) * (1 - x^n)^(n-2). (I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n+1)*n*(n-1)*(n-2)*(n-3)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2). LINKS Paul D. Hanna, Table of n, a(n) for n = 4..2050 FORMULA G.f. A(x) = Sum_{n>=4} a(n)*x^n satisfies: (1) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2). (2) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(5*n) * (1 - x^n)^(n-2). (3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)*(n+4)/120 * x^(4*n) * (1 - x^n)^(n-2). (4) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-3)) / (1 - x^n)^(n+2). (5) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-2)) / (1 - x^n)^(n+2). (6) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)*(n-3)*(n-4)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2). EXAMPLE G.f.: A(x) = x^4 + x^5 + x^6 + x^7 + 7*x^8 + x^9 + x^10 + x^11 + 22*x^12 + x^13 + x^14 - 19*x^15 + 57*x^16 + x^17 + x^18 + x^19 + 22*x^20 + ... where A(x) = ... + 5*x^(-20)*(1 - x^(-5))^(-7) + 1*x^(-16)*(1 - x^(-4))^(-6) + 0*x^(-12) + 0*x^(-8) + 0*x^(-4) + 0 + 1*x^4/(1-x) + 5*x^8 + 15*x^12*(1 - x^3) + 35*x^16*(1 - x^4)^2 + 70*x^20*(1 - x^5)^3 + ... + n*(n+1)*(n+2)*(n+3)/24 * x^(4*n)*(1 - x^n)^(n-2) + ... PROG (PARI) {a(n) = my(A = sum(m=-n-1, n+1, if(m==0, 0, m*(m+1)*(m+2)*(m+3)/24 * x^(4*m) * (1 - x^m +x*O(x^n))^(m-2) )) ); polcoeff(A, n)} for(n=4, 100, print1(a(n), ", ")) CROSSREFS Cf. A291937, A356774, A356775, A357156. Sequence in context: A295294 A317936 A277875 * A317935 A360439 A115064 Adjacent sequences: A357154 A357155 A357156 * A357158 A357159 A357160 KEYWORD sign AUTHOR Paul D. Hanna, Sep 22 2022 STATUS approved

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Last modified September 8 12:12 EDT 2024. Contains 375753 sequences. (Running on oeis4.)