A272131 a(n) = 384*n^3 - 1184*n^2 + 1228*n - 427.

-427, 1, 365, 2969, 10117, 24113, 47261, 81865, 130229, 194657, 277453, 380921, 507365, 659089, 838397, 1047593, 1288981, 1564865, 1877549, 2229337, 2622533, 3059441, 3542365, 4073609, 4655477, 5290273, 5980301, 6727865, 7535269, 8404817, 9338813, 10339561

Offset: 0

Author

Vincenzo Librandi, Apr 26 2016

Keywords

• sign
• 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).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A272129, A272132, A245244.

Programs

Magma

[384*n^3 - 1184*n^2 + 1228*n - 427: n in [0..50]];

Maple

[384*n^3-1184*n^2+1228*n-427$n=0..35]; # Muniru A Asiru, Jan 28 2019

Mathematica

Table[384 n^3 - 1184 n^2 + 1228 n - 427, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{-427,1,365,2969},40] (* Harvey P. Dale, Aug 24 2024 *)

PARI

lista(nn) = for(n=0, nn, print1(384*n^3-1184*n^2+1228*n-427, ", ")); \\ Altug Alkan, Apr 26 2016

Formula

O.g.f.: (-427 + 1709*x - 2201*x^2 + 3223*x^3)/(1-x)^4.

E.g.f.: (-427 + 428*x - 32*x^2 + 384*x^3)*exp(x).

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) = (2*n-1)^2*A272129(n) - 4*(n-1)^2*A272129(n-1).

A272132(n) = (2*n-1)^2*a(n) - 4*(n-1)^2*a(n-1).

n*a(n) = 1 + 3^7*(n-1)/(n+1) + 5^7*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019