A185375 a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.

0, 0, 1, 91, 966, 5082, 18447, 53053, 129948, 282948, 562989, 1043119, 1824130, 3040830, 4868955, 7532721, 11313016, 16556232, 23683737, 33201987, 45713278, 61927138, 82672359, 108909669, 141745044, 182443660

Offset: 0

Author

Wesley Transue, Jan 21 2012

Keywords

• easy
• nonn

Comments

Third column (k=2) of A008958.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Third column (k=2) of A008958 Triangle of central factorial numbers.

Cf. A103220.

Programs

Magma

[n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90 : n in [0..50]]; // Wesley Ivan Hurt, Apr 23 2021

Mathematica

Table[n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n - 17)/90, {n, 0, 50}] (* G. C. Greubel, Jun 28 2017 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,1,91,966,5082,18447},30] (* Harvey P. Dale, Oct 10 2021 *)

PARI

a(n) = binomial(2*n+1,5)*(10*n-17)/3  \\ Michel Marcus, Jun 18 2013

Formula

a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.

a(n) = binomial(2*n+1,5)*(10*n-17)/3.

From G. C. Greubel, Jun 28 2017: (Start)

G.f.: x^2*(1 + 84*x + 350*x^2 + 196*x^3 + 9*x^4)/(1 - x)^7.

E.g.f.: (1/90)*x^2*(45 + 1320 x + 2280 x^2 + 864 x^3 + 80 x^4)*exp(x). (End)

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Apr 23 2021

a(n) = Sum_{1 <= i <= j <= n-1} (2*i - 1)^2 * (2*j - 1)^2. - Peter Bala, Sep 03 2023