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 A354647 G.f. A(x) satisfies: -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2). 1
 1, 0, 1, 3, 9, 25, 78, 256, 881, 3064, 10831, 38766, 140550, 514625, 1900301, 7067013, 26448613, 99539716, 376489459, 1430330451, 5455742957, 20885223619, 80213926069, 309002022843, 1193616950854, 4622372591972, 17942238661229, 69795082381496, 272046051362013 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..28. FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: (1) -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2). (2) -x^2 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2). (3) -x^2 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2). (4) -x^2 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity. EXAMPLE G.f.: A(x) = 1 + x^2 + 3*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 256*x^7 + 881*x^8 + 3064*x^9 + 10831*x^10 + 38766*x^11 + 140550*x^12 + ... such that A = A(x) satisfies: (1) -x^2 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ... (2) -x^2 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ... (3) -x^2 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ... (4) -x^2 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(x^2 + sum(m=0, sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ), #A-1) ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A268650, A354648, A354649. Sequence in context: A212352 A198180 A101786 * A217995 A246653 A192371 Adjacent sequences: A354644 A354645 A354646 * A354648 A354649 A354650 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 21 2022 STATUS approved

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Last modified June 10 14:31 EDT 2023. Contains 363205 sequences. (Running on oeis4.)