A331528 a(n) = n^2 * (n+1)^2 * (n^2+n+1) / 12.

0, 1, 21, 156, 700, 2325, 6321, 14896, 31536, 61425, 111925, 193116, 318396, 505141, 775425, 1156800, 1683136, 2395521, 3343221, 4584700, 6188700, 8235381, 10817521, 14041776, 18030000, 22920625, 28870101, 36054396, 44670556, 54938325, 67101825, 81431296, 98224896

Offset: 0

Author

Seiichi Manyama, Jan 19 2020

Keywords

• nonn
• easy

Comments

Let b(n,k) = Sum_{j=0..n} (-1)^(n-j)* j^k * binomial(n,j) * binomial(n+j,j).

b(n,0) = 1.

b(n,1) = 1/1! * n * (n+1).

b(n,2) = 1/2! * n^2 * (n+1)^2.

b(n,3) = 1/3! * n^2 * (n+1)^2 * (n^2+n+1) (= 2*a(n)).

b(n,4) = 1/4! * n^3 * (n+1)^3 * (n^2+n+4).

b(n,5) = 1/5! * n^2 * (n+1)^2 * (n^6+3*n^5+13*n^4+21*n^3+18*n^2+8*n-4).

b(n,6) = 1/6! * n^3 * (n+1)^3 * (n^2+n+4) * (n^4+2*n^3+17*n^2+16*n-6).

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A002378 (b(n,1)), A163102 (b(n,2)), A168178 (first differences).

Programs

Magma

[n^2*(n+1)^2*(n^2+n+1)/12:n in [0..32]]; // Marius A. Burtea, Jan 19 2020

Magma

R:=PowerSeriesRing(Integers(), 33); [0] cat (Coefficients(R!( x*(x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7))); // Marius A. Burtea, Jan 19 2020

Mathematica

a[n_] := (n*(n+1))^2 * (n^2+n+1) / 12; Array[a, 33, 0] (* Amiram Eldar, May 05 2021 *)

PARI

{a(n) = n^2*(n+1)^2*(n^2+n+1)/12}

Formula

G.f.: x * (x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7.