A346943 a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.

1, 1, 7, 19, 159, 729, 7407, 48231, 581535, 4922325, 68891175, 718638075, 11465661375, 142257791025, 2550046679775, 36691916525775, 730304613424575, 11958031070311725, 261722208861516375, 4805774015579971875, 114729101737416849375, 2334996696935363855625

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).

a(n) ~ (Pi - 2) * n^(n+2) / exp(n).

E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.

E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).

Mathematica

RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n, 0, 20}]
nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!

Crossrefs

Cf. A000246, A000932, A024167.

Sequence in context: A328713 A096321 A201806 * A128338 A173384 A107195

Adjacent sequences: A346940 A346941 A346942 * A346944 A346945 A346946

Keyword

nonn

Author

Vaclav Kotesovec, Aug 08 2021, following a suggestion from John M. Campbell

Status

approved