A272133 a(n) = 122880*n^5 - 829440*n^4 + 2258688*n^3 - 3076288*n^2 + 2079892*n - 555731.

-555731, 1, 29525, 1657129, 16591741, 80872529, 269614501, 711754105, 1604794829, 3229552801, 5964902389, 10302521801, 16861638685, 26403775729, 39847496261, 58283149849, 82987617901, 115439059265, 157331655829, 210590358121, 277385630909, 360148198801

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 (6,-15,20,-15,6,-1).

Crossrefs

Cf. A272132, A245244.

Programs

Magma

[122880*n^5 - 829440*n^4 + 2258688*n^3 -3076288*n^2 + 2079892*n - 555731: n in [0..30]];

Maple

[122880*n^5-829440*n^4+2258688*n^3-3076288*n^2+2079892*n-555731$n=0..30]; # Muniru A Asiru, Jan 28 2019

Mathematica

Table[122880 n^5 - 829440 n^4 + 2258688 n^3 - 3076288 n^2 + 2079892 n - 555731, {n, 0, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{-555731,1,29525,1657129,16591741,80872529},30] (* Harvey P. Dale, Feb 10 2021 *)

PARI

lista(nn) = for(n=0, nn, print1(122880*n^5 - 829440*n^4 + 2258688*n^3 - 3076288*n^2 + 2079892*n - 555731, ", ")); \\ Altug Alkan, Apr 26 2016

Formula

O.g.f.: (-555731 + 3334387*x - 8306446*x^2 + 12594614*x^3 - 1244143*x^4 + 8922919*x^5)/(1-x)^6.

E.g.f.: (-555731 + 555732*x - 263104*x^2 + 354048*x^3 + 399360*x^4 + 122880*x^5)*exp(x).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (2*n-1)^2*A272132(n) - 4*(n-1)^2*A272132(n-1), see page 7 in Brent's paper.

n*a(n) = 1 + 3^11*(n-1)/(n+1) + 5^11*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019