A035028 First differences of A002002.

4, 20, 104, 552, 2972, 16172, 88720, 489872, 2719028, 15157188, 84799992, 475894200, 2677788492, 15102309468, 85347160608, 483183316512, 2739851422820, 15558315261812, 88462135512712, 503569008273992, 2869602773253884, 16368396446913420, 93449566652932784

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001850, A002002, A035029, A049600.

Programs

Magma

I:=[4,20,104]; [n le 3 select I[n] else ( (7*n+1)*Self(n-1) - (7*n-15)*Self(n-2) + (n-3)*Self(n-3) )/(n+1): n in [1..30]]; // G. C. Greubel, Oct 19 2022

Mathematica

Differences[CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}], x]] (* Harvey P. Dale, Jun 04 2011 *)
With[{P=LegendreP}, Table[(n*(n+2)*P[n+2,3] -(6*n^2+10*n+1)*P[n+1,3] +(n+1)*(5*n+ 3)*P[n,3])/(2*n*(n+1)), {n,30}]] (* G. C. Greubel, Oct 19 2022 *)

SageMath

def A001850(n): return gen_legendre_P(n,0,3)
def A035028(n): return ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(2*(n+1)*(n+2))
[A035028(n) for n in range(40)] # G. C. Greubel, Oct 19 2022

Formula

a(n) = A049600(n, n-3).

D-finite with recurrence: (n+2)*a(n) - (7*n+8)*a(n-1) + (7*n-8)*a(n-2) - (n-2)*a(n-3) = 0. - R. J. Mathar, Jan 28 2020

a(n) = ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(2*(n+1)*(n+2)), A001850(n) = LegrndreP(n, 3). - G. C. Greubel, Oct 19 2022

Extensions

More terms from Harvey P. Dale, Jun 04 2011