A269450 a(n) = (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3) with a(0)=0, a(1)=a(2)=1.

0, 1, 1, 1, 4, 18, 90, 542, 3812, 30584, 275780, 2761524, 30406824, 365153944, 4749732736, 66526393072, 998258318560, 15976852694928, 271672659784368, 4891101414839216, 92946837570669440, 1859207430534452576, 39048231231949618976, 859153763261341223136

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) = 0.7651976865579665514497175261... . - Vaclav Kotesovec, Feb 27 2016

Example

G.f. = x + x^2 + x^3 + 4*x^4 + 18*x^5 + 90*x^6 + 542*x^7 + 3812*x^8 + ...

Mathematica

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

Prog

(PARI) {a(n) = if( n<3, n>0, (n-1)*a(n-1) - a(n-2) + (n-2)*a(n-3))};
(PARI) {a(n) = my(A); if( n<3, n>0, A = vector(n, i, 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, 1, 1]; [0] 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. A269509, A269820.

Sequence in context: A011270 A367724 A355247 * A206639 A367875 A172964

Adjacent sequences: A269447 A269448 A269449 * A269451 A269452 A269453

Keyword

nonn,easy

Author

Michael Somos, Feb 27 2016

Status

approved