OFFSET
0,1
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = A035343(n+2, 5), n >= 0 (sixth column of quintinomial coefficients).
a(n) = A062750(n+2, 5), n >= 0 (sixth column).
G.f.: (x^2)*(2-x)*(2 - 2*x + x^2)/(1-x)^6. (For numerator polynomial see N5(5, x) = 4 - 6*x + 4*x^2 - x^3 from A063422.)
a(n) = binomial(n+6, 5) - binomial(n+2, 1). - Zerinvary Lajos, May 08 2006
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), with a(0)=4, a(1)=18, a(2)=52, a(3)=121, a(4)=246, a(5)=455. - Harvey P. Dale, Aug 18 2012
From G. C. Greubel, Aug 01 2022: (Start)
a(n) = Sum_{j=0..3} binomial(n+j+2, j+2).
E.g.f.: (1/120)*(480 +1680*x +1200*x^2 +300*x^3 +30*x^4 +x^5)*exp(x). (End)
MAPLE
seq(1/120*(n+8)*(n+2)*(n+1)*(n^2+9*n+30), n=0..40);
MATHEMATICA
Table[Sum[Binomial[n+i, i], {i, 2, 5}], {n, 0, 30}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {4, 18, 52, 121, 246, 455}, 30] (* Harvey P. Dale, Aug 18 2012 *)
Sum[(-1)^j*Binomial[4*j-2 + Range[0, 60], 4*j-3], {j, 2}] (* G. C. Greubel, Aug 01 2022 *)
PROG
(PARI) a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [Binomial(n+6, 5) -(n+2): n in [0..60]]; // G. C. Greubel, Aug 01 2022
(SageMath) [binomial(n+6, 5) -(n+2) for n in (0..60)] # G. C. Greubel, Aug 01 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved