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 A357795 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2). 3
 1, 4, 26, 300, 4134, 61696, 969660, 15837400, 266125823, 4571229248, 79904206064, 1416736880104, 25418030469904, 460600399886240, 8417980252615072, 154985730303047328, 2871904782258356719, 53519211809275995362, 1002383232008661189884, 18858606600633628740774 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (y - x^n)^(n-1), which holds formally for all y. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following. (1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2). (2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)/3! * x^(n*(n-3)) / ((1 - x^(n-2))^n * A(x)^(n-2)). EXAMPLE G.f.: A(x) = 1 + 4*x + 26*x^2 + 300*x^3 + 4134*x^4 + 61696*x^5 + 969660*x^6 + 15837400*x^7 + 266125823*x^8 + 4571229248*x^9 + 79904206064*x^10 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(n=-#A-1, #A+1, n*(n+1)*(n+2)/3! * x^n * if(n==-2, 0, (1 - x^(n+2) +x*O(x^#A) )^n) * Ser(A)^(n+2) ), #A-1) ); H=A; 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( sum(n=-#A-1, #A+1, (-1)^(n-1) * n*(n-1)*(n-2)/3! * x^(n*(n-3)) * if(n==2, 0, 1/(1 - x^(n-2) +x*O(x^#A) )^n) / Ser(A)^(n-2) ), #A-1) ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A357158, A357794, A357796. Sequence in context: A167811 A156306 A054592 * A102202 A271613 A271614 Adjacent sequences: A357792 A357793 A357794 * A357796 A357797 A357798 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 22 2022 STATUS approved

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Last modified August 12 19:26 EDT 2024. Contains 375113 sequences. (Running on oeis4.)