A272379 a(n) = n*(24*n^3 - 60*n^2 + 54*n - 17).

0, 1, 86, 759, 3100, 8765, 19986, 39571, 70904, 117945, 185230, 277871, 401556, 562549, 767690, 1024395, 1340656, 1725041, 2186694, 2735335, 3381260, 4135341, 5009026, 6014339, 7163880, 8470825, 9948926, 11612511, 13476484, 15556325, 17868090, 20428411

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 (5,-10,10,-5,1)

Crossrefs

Cf. A272378, A272380.

Programs

Magma

[n*(24*n^3 - 60*n^2 + 54*n - 17): n in [0..50]];

Mathematica

Table[n (24 n^3 - 60 n^2 + 54 n - 17), {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,86,759,3100},40] (* Harvey P. Dale, Mar 24 2021 *)

PARI

vector(100, n, n--; n*(24*n^3 - 60*n^2 + 54*n - 17)) \\ Altug Alkan, Apr 29 2016

Formula

G.f.: x*(1 + 81*x + 339*x^2 + 155*x^3)/(1 - x)^5.

E.g.f.: x*(1 + 42*x + 84*x^2 + 24*x^3)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

See page 7 in Brent's paper:

a(n) = n^2*A272378(n) - n*(n-1)*A272378(n-1),

A272380(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*(24*x^3 - 60*x^2 + 54*x - 17) = Sum_{n >= 0} ((n+1)^9 + n^9)*a(n,x) and x*(24*x^3 - 60*x^2 + 54*x - 17) = Sum_{n >= 0} ((n+1)^8 - n^8)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 9/2. See the Bala link in A036970. Cf. A272378 and A272380. (End)