A272378 a(n) = n*(6*n^2 - 8*n + 3).

0, 1, 22, 99, 268, 565, 1026, 1687, 2584, 3753, 5230, 7051, 9252, 11869, 14938, 18495, 22576, 27217, 32454, 38323, 44860, 52101, 60082, 68839, 78408, 88825, 100126, 112347, 125524, 139693, 154890, 171151, 188512, 207009, 226678, 247555, 269676, 293077

Offset: 0

Author

Vincenzo Librandi, Apr 29 2016

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
• Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Crossrefs

Cf. A000384, A015237, A272379, A272380.

Programs

Magma

[n*(6*n^2 - 8*n + 3): n in [0..50]];

Mathematica

Table[n (6 n^2 - 8 n + 3), {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,1,22,99},40] (* Harvey P. Dale, Dec 29 2017 *)

PARI

vector(100, n, n--; n*(6*n^2 - 8*n + 3)) \\ Altug Alkan, Apr 29 2016

Python

for n in range(0,10**3):print(n*(6*n**2-8*n+3),end=", ") # Soumil Mandal, Apr 30 2016

Formula

G.f.: x*(1 + 18*x + 17*x^2)/(1 - x)^4.

E.g.f.: x*(1 + 10*x + 6*x^2)*exp(x).

a(n) = n*A080859(n+1).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3.

See page 7 in Brent's paper:

a(n) = n^2*A000384(n) - n*(n-1)*A000384(n-1).

A272379(n) = n^2*a(n) - n*(n-1)*a(n-1).

From Peter Bala, Jan 30 2019: (Start)

Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have x^2*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^7 + n^7)*a(n,x) and x*(6*x^2 - 8*x + 3) = Sum_{n >= 0} ((n+1)^6 - n^6)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 7/2. See the Bala link in A036970. Cf. A272379. (End)