A343446 Coefficients of the series S(p, q) for which -(p^(1/3))*S converges to the largest real root of x^4 - p*x + q, where 0 < p and 0 < q < 3*(p/4)^(4/3).
-1, 1, 4, 40, 648, 14560, 418880, 14696640, 608608000, 29056867840, 1571364748800, 94937979136000, 6337884013260800, 463301182536192000, 36806315255277568000, 3157533815406530560000, 290912372128665391104000, 28648563542097847828480000
Offset: 0
Links
- Albert Eagle, Series for all the roots of a trinomial equation, Am. Math. Monthly, vol. 46, no. 7 (Aug. - Sep., 1939), pp. 422 - 425.
Crossrefs
Programs
-
Maple
a := proc(n) option remember; if n < 3 then [-1, 1, 4][n+1] else 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) fi; end: seq(a(n), n = 0..20); # Peter Bala, Jul 23 2024
-
Mathematica
Clear[a]; a=Table[3^(n - 1) Pochhammer[(n + 2)/3, n - 1], {n, 0, 20}] (* In general, for the series S(m, p, q) for which (-p^(1/m))*S converges to the largest real root of x^(m + 1) - p*x + q, the first n + 1 coefficients are: *) Clear[c]; c[m_,n_] := Table[m^(k - 1) Pochhammer[(k + m - 1)/m, k - 1], {k, 0, n}](* and S(m, p, q) to n + 1 terms is given by *) Clear[s]; s[m_,p_,q_,n_]:= Sum[c[m,n][[k + 1]]*q^k/((m^k)*(p^(k (m + 1)/m))*k!), {k, 0, n}]
Formula
a(n) = 3^(n - 1)*((n + 2)/3)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(x).
a(n) = 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) with a(0) = -1, a(1) = 1 and a(2) = 4. - Peter Bala, Jul 23 2024
Comments