A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).

4, 18, 52, 121, 246, 455, 784, 1278, 1992, 2992, 4356, 6175, 8554, 11613, 15488, 20332, 26316, 33630, 42484, 53109, 65758, 80707, 98256, 118730, 142480, 169884, 201348, 237307, 278226, 324601, 376960, 435864, 501908, 575722, 657972, 749361, 850630, 962559

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A035343, A062750, A063422.

Partial sums of A063258.

Programs

Magma

[Binomial(n+6, 5) -(n+2): n in [0..60]]; // G. C. Greubel, Aug 01 2022

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 *)

PARI

a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[binomial(n+6, 5) -(n+2) for n in (0..60)] # G. C. Greubel, Aug 01 2022

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)