OFFSET
0,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
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)*(+2*a(n+1) - a(n+2) + 4*a(n+3) - a(n+4)) + a(n+1)*(+a(n+1) + 2*a(n+3)) + a(n+2)*(-a(n+2) + 2*a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n=>0.
a(n) ~ c * 2^n * (n-1)!, where c = (BesselJ(0, 1/2) - BesselJ(1, 1/2))/2 = 0.348100674782969508922225... . - Vaclav Kotesovec, Mar 07 2016
EXAMPLE
G.f. = 1 + x + x^2 + 5*x^3 + 33*x^4 + 265*x^5 + 2657*x^6 + 31949*x^7 + ...
MATHEMATICA
RecurrenceTable[{a[n] == 2*(n-1)*a[n-1] - a[n-2] + 2*(n-2)*a[n-3], a[0]==a[1]==a[2]==1}, a, {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2016 *)
FullSimplify[Assuming[Element[x, Reals], CoefficientList[Series[Pi/4 *((BesselJ[1, 1/2] - BesselJ[0, 1/2]) * BesselY[0, -1/2 + x] + BesselJ[0, 1/2 - x] * (BesselY[0, -1/2] + BesselY[1, -1/2])), {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Mar 07 2016 *)
PROG
(PARI) {a(n) = if( n<3, n>=0, 2*(n-1)*a(n-1) - a(n-2) + 2*(n-2)*a(n-3))};
(PARI) {a(n) = my(A); if( n<3, n>=0, A = vector(n, i, 1); A[3]=5; for(i=4, n, A[i] = 2*(i-1)*A[i-1] - A[i-2] + 2*(i-2)*A[i-3]); A[n])};
(Magma) I:=[1, 1, 5]; [1] cat [n le 3 select I[n] else 2*(n-1)*Self(n-1) - Self(n-2) + 2*(n - 2)*Self(n-3): n in [1..50]]; // G. C. Greubel, Jul 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 07 2016
STATUS
approved