A357795 - OEIS

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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).

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^(3n) (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 + 4x + 26x^2 + 300x^3 + 4134x^4 + 61696x^5 + 969660x^6 + 15837400x^7 + 266125823x^8 + 4571229248x^9 + 79904206064x^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) +xO(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 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)