OFFSET
1,2
FORMULA
E.g.f. A(x) satisfies the differential equation 6*A(x) + (6*x + 2)*A'(x) + (x^2 - 2)*A''(x) = 0, A(0) = 1, A'(0) = 1.
a(n) ~ n! * c * n^(1 + 1/sqrt(2)) / 2^(n/2), where c = 0.42906449224324091038170340685604072807700713285504473...
MATHEMATICA
RecurrenceTable[{a[1]==1, a[2]==4, a[n]==a[n-1] + n*(n+1)/2*a[n-2]}, a, {n, 1, 20}]
nmax = 20; Round[Rest[CoefficientList[Series[(Sqrt[2]*(2 + x^2) - 4*x) * (((2 - x^2)^2 * (1 - 3*Sqrt[2]) * Hypergeometric2F1[2, 3, 3 - 1/Sqrt[2], (2 + Sqrt[2]*x)/4] + 16*(1 + x) * ((Sqrt[2] + x)/(Sqrt[2] - x))^(1/ Sqrt[2]) * ((1 - 3*Sqrt[2]) * Hypergeometric2F1[2, -1/Sqrt[2], 3 - 1/Sqrt[2], -1] + 6*Hypergeometric2F1[3, -1/Sqrt[2], 4 - 1/Sqrt[2], -1])) / (8*(Sqrt[2] - x)^2 * (2 - x^2)^2 * (3*Sqrt[2] * Hypergeometric2F1[3, -1/Sqrt[2], 4 - 1/Sqrt[2], -1] - (6 - Sqrt[2]) * Hypergeometric2F1[2, -1/Sqrt[2], 3 - 1/Sqrt[2], -1]))), {x, 0, nmax}], x]] * Range[nmax]!]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jan 30 2022, following a suggestion from John M. Campbell
STATUS
approved