A269509 a(n) = (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3) with a(0)=a(1)=1, a(2)=0.

1, 1, 0, 0, 2, 8, 38, 230, 1620, 12996, 117184, 1173424, 12920440, 155160880, 2018252088, 28268334072, 424179011312, 6788869628240, 115438898013920, 2078322337814624, 39494885173772256, 790012720199894976, 16592338685780314720, 365070830955616246240

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

E.g.f. A(x) satisfies 0 = A'(x)*(A'(x) + A'''(x)) - (A(x) + A''(x))*(A(x) + 2*A''(x)).

0 = a(n)*(+a(n+1) - a(n+2) + 2*a(n+3) - a(n+4)) + a(n+1)*(+a(n+1) + 2*a(n+3)) + a(n+2)*(-a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0.

a(n) ~ c * (n-1)!, where c = BesselJ(0,1) - BesselJ(1,1) = 0.3251471008130330354900353223837483... . - Vaclav Kotesovec, Feb 29 2016

Example

G.f. = 1 + x + 2*x^4 + 8*x^5 + 38*x^6 + 230*x^7 + 1620*x^8 + 12996*x^9 + ...

Mathematica

RecurrenceTable[{a[n] == (n-1)*a[n-1] - a[n-2] + (n-2)*a[n-3], a[0]==a[1]==1, a[2]==0}, a, {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2016 *)
FullSimplify[Assuming[Element[x, Reals], CoefficientList[Series[Pi/2 * ((BesselJ[1, 1] - BesselJ[0, 1]) * BesselY[0, -1 + x] + BesselJ[0, 1 - x] * (BesselY[0, -1] + BesselY[1, -1])), {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Mar 07 2016 *)

Prog

(PARI) {a(n) = if( n<3, n>=0&&n<2, (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3))};
(PARI) {a(n) = my(A); if( n<3, n>=0&&n<2, A = vector(n, i); A[1]=1; for(i=4, n, A[i] = (i-1)*A[i-1] - A[i-2] + (i-2)*A[i-3]); A[n])};
(Magma) I:=[1, 0, 0]; [1] cat [n le 3 select I[n] else (n-1)*Self(n-1) - Self(n-2) + (n - 2)*Self(n-3): n in [1..50]]; // G. C. Greubel, Jul 30 2018

Crossrefs

Cf. A269450, A269820.

Sequence in context: A058786 A096654 A371312 * A307725 A308205 A191016

Adjacent sequences: A269506 A269507 A269508 * A269510 A269511 A269512

Keyword

nonn,easy

Author

Michael Somos, Feb 28 2016

Status

approved