A117651 A002415 and A052472 interlaced.

1, 0, 2, 1, 0, 6, 10, 20, 35, 50, 84, 105, 168, 196, 300, 336, 495, 540, 770, 825, 1144, 1210, 1638, 1716, 2275, 2366, 3080, 3185, 4080, 4200, 5304, 5440, 6783, 6936, 8550, 8721, 10640, 10830, 13090, 13300, 15939, 16170, 19228, 19481, 23000, 23276, 27300

Offset: 0

Author

Roger L. Bagula, Apr 11 2006

Keywords

• nonn
• less
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Crossrefs

Cf. A002415, A052472, A018226.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x -2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/( (1+x)^4*(1-x)^5) )); // G. C. Greubel, May 19 2019

Mathematica

f[n_]:= n*(n+1)*(n+2)*(n-3)/12; g[n_]:= n^2*(n^2 -1)/12; Table[{Abs[f[n]], g[n]}, {n, 1, 25}]//Flatten
LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1}, {1,0,2,1,0,6,10,20,35,50,84, 105}, 50] (* Harvey P. Dale, Mar 05 2016 *)

PARI

my(x='x+O('x^50)); Vec((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10-4*x^11 )/((1+x)^4*(1-x)^5)) \\ G. C. Greubel, May 19 2019

Sage

((1-x-2*x^2+3*x^3-3*x^4+4*x^5+16*x^6-16*x^7 -14*x^8+14*x^9+4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, May 19 2019

Formula

G.f.: (1 -x -2*x^2 +3*x^3 -3*x^4 +4*x^5 +16*x^6 -16*x^7 -14*x^8 +14*x^9 +4*x^10 -4*x^11 )/((1+x)^4*(1-x)^5). - Colin Barker, Mar 15 2013

a(n) = abs((2*n^4 +12*n^3 -2*n^2 -132*n -195 +(4*n^3 -6*n^2 -124*n -189)*(-1)^n))/384. - Luce ETIENNE, Jun 01 2015

a(n) = abs((-3*(65 +63*(-1)^n) -4*(33 +31*(-1)^n)*n -2*(1+3*(-1)^n)*n^2 +4*(3 +(-1)^n)*n^3 +2*n^4)/384). - Colin Barker, Jun 02 2015

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 11. - Charles R Greathouse IV, Jun 02 2015