A351531 a(0)=1; a(1)=1; for n>1, a(n) = a(n-1) + 3*n*a(n-2).

1, 1, 7, 16, 100, 340, 2140, 9280, 60640, 311200, 2130400, 12400000, 89094400, 572694400, 4314659200, 30085907200, 237189548800, 1771570816000, 14579806451200, 115559342963200, 990347730035200, 8270586336716800, 73633536519040000, 644303993752499200, 5945918623123379200

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

E.g.f. A(x) satisfies the differential equation 6*A(x) + (3*x + 1)*A'(x) - A''(x) = 0, A(0) = 1, A'(0) = 1.

E.g.f.: 1 + sqrt(Pi/6) * (1 + 3*x) * exp((1 + 3*x)^2/6) * (erf((1 + 3*x)/sqrt(6)) - erf(1/sqrt(6))).

a(n) ~ erfc(1/sqrt(6)) * sqrt(Pi) * 3^(n/2) * exp(sqrt(n/3) - n/2 + 1/12) * n^((n+1)/2) / 2 * (1 + 55/(72*sqrt(3*n)) + 7561/(31104*n)).

Mathematica

RecurrenceTable[{a[0]==1, a[1]==1, a[n]==a[n-1] + 3*n*a[n-2]}, a, {n, 0, 20}]
nmax = 20; FullSimplify[CoefficientList[Series[1 + Sqrt[Pi/6] * (1 + 3*x) * E^((1 + 3*x)^2/6) * (Erf[(1 + 3*x)/Sqrt[6]] - Erf[1/Sqrt[6]]), {x, 0, nmax}], x] * Range[0, nmax]!]

Crossrefs

Sequence in context: A363173 A214904 A029498 * A181777 A332573 A129422

Adjacent sequences: A351528 A351529 A351530 * A351532 A351533 A351534

Keyword

nonn,easy

Author

Vaclav Kotesovec, Feb 13 2022, following a suggestion from John M. Campbell

Status

approved