A244864 a(n) = binomial(n+5,5) + 4*binomial(n+4,5) + 4*binomial(n+3,5) + binomial(n+2,5).

1, 10, 49, 165, 440, 1001, 2030, 3774, 6555, 10780, 16951, 25675, 37674, 53795, 75020, 102476, 137445, 181374, 235885, 302785, 384076, 481965, 598874, 737450, 900575, 1091376, 1313235, 1569799, 1864990, 2203015, 2588376, 3025880, 3520649, 4078130, 4704105, 5404701, 6186400, 7056049

Offset: 0

Author

N. J. A. Sloane, Jul 07 2014

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Programs

Maple

a:= n-> (2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 11 2014

Mathematica

Table[Binomial[n+5,5]+4*Binomial[n+4,5]+4*Binomial[n+3,5]+ Binomial[ n+2,5],{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,49,165,440,1001},40] (* Harvey P. Dale, Nov 13 2014 *)
a[n_] := (2 n^5 + 15 n^4 + 48 n^3 + 81 n^2 + 70 n + 24)/  24; Array[a, 40, 0] (* or *)
CoefficientList[Series[(x^3 + 4 x^2 + 4 x + 1)/(x - 1)^6, {x, 0, 40}], x] (* Robert G. Wilson v, Feb 26 2015 *)

PARI

a(n)=(2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

Formula

G.f.: (x+1)*(x^2+3*x+1)/(x-1)^6; a(n) = (2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24. - Alois P. Heinz, Jul 11 2014

a(n) = Sum_{k=A000292(n)..A000292(n+1)} k. - J. M. Bergot, Feb 25 2015