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 A354653 G.f. A(x) satisfies: -3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2). 9
 4, 63, 3024, 188688, 13492350, 1044853344, 85281392688, 7224776707896, 629288553814092, 56002675660109424, 5070000855941708292, 465454828626459320736, 43230859988456631732954, 4054827527508982869148392, 383529048423080768494135488, 36541031890621600233033859488 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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: (1) -3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2). (2) -3 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2). (3) -3 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2). (4) -3 = 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. a(n) = (-1)^(n+1) * Sum_{k=0..2*n+1} A354649(n,k)*(-3)^k, for n >= 0. a(n) = Sum_{k=0..2*n+1} A354650(n,k)*3^k, for n >= 0. EXAMPLE G.f.: A(x) = 4 + 63*x + 3024*x^2 + 188688*x^3 + 13492350*x^4 + 1044853344*x^5 + 85281392688*x^6 + 7224776707896*x^7 + 629288553814092*x^8 + ... such that A = A(x) satisfies: (1) -3 = ... + 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) -3 = (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) -3 = (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) -3 = (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=[4]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(3 + 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. A354649, A354650, A268299, A354652, A354654, A354661, A354662, A354663, A354664. Sequence in context: A292394 A351779 A203512 * A213545 A362386 A361683 Adjacent sequences: A354650 A354651 A354652 * A354654 A354655 A354656 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 02 2022 STATUS approved

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Last modified June 18 03:29 EDT 2024. Contains 373468 sequences. (Running on oeis4.)