A317346 O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n^2*A(x) ) = 0 for n >= 1.
1, 2, 72, 8096, 1839000, 695334816, 392764566208, 309340607492096, 323795915817507936, 434750954619876448000, 728547799352068864173632, 1490865523016798790557180928, 3659466509860384349989504297344, 10614823215131644149237135937187328, 35927108634064565449228268842108588800, 140351379904337650357154561973550135705600
Offset: 1
Keywords
Examples
O.g.f.: A(x) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ... such that [x^n] exp( n^3*x - n^2*A(x) ) = 0 for n >= 1. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k/k! in exp( n^3*x - n^2*A(x) ) begins: n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...]; n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...]; n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...]; n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...]; n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...]; n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...]; n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ... in which the coefficient of x^n in row n forms a diagonal of zeros. RELATED SERIES. exp(A(x)) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + ... + A317345(n)*x^n/n! + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..200
Programs
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PARI
{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); polcoeff( log(Ser(A)),n)} for(n=1,20,print1(a(n),", "))
Formula
a(n) ~ sqrt(1 - c) * 3^(3*n - 7/3) * n^(2*n - 5/2) / (sqrt(2*Pi) * exp(2*n) * c^(n - 1/3) * (3 - c)^(2*n - 2)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 07 2018
Comments