A027622 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5.

1114, 3413, 8476, 18247, 35414, 63529, 107128, 171851, 264562, 393469, 568244, 800143, 1102126, 1488977, 1977424, 2586259, 3336458, 4251301, 5356492, 6680279, 8253574, 10110073, 12286376, 14822107, 17760034, 21146189, 25029988, 29464351, 34505822, 40214689

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Patrick De Geest, Palindromic Quasi_Under_Squares of the form n+(n+1)^2
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000027, A027620, A027621, A028387.

Programs

Magma

[n+(n+1)^2+(n+2)^3+(n+3)^4+(n+4)^5: n in [0..30]]; // Vincenzo Librandi, Dec 28 2010

Maple

seq( add((n+j)^(j+1), j=0..4), n=0..30); # G. C. Greubel, Aug 05 2022

Mathematica

Table[n +(n+1)^2 +(n+2)^3 +(n+3)^4 +(n+4)^5, {n, 0, 29}] (* Alonso del Arte, Nov 22 2016 *)
Table[ReleaseHold@ Total@ MapIndexed[#1^First@ #2 &, Rest@ FactorList[ Pochhammer[Hold@ n, 5]][[All, 1]]], {n, 0, 29}] (* or *)
CoefficientList[Series[(1114 -3271x +4708x^2 -3694x^3 +1522x^4 -259x^5)/(1-x)^6, {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)
Table[Total[Table[(n+k)^(k+1),{k,0,4}]],{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {1114,3413,8476,18247,35414,63529}, 30] (* Harvey P. Dale, Aug 04 2022 *)

PARI

Vec((1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6 + O(x^30)) \\ Colin Barker, Dec 05 2016

SageMath

[sum((n+j)^(j+1) for j in (0..4)) for n in (0..30)] # G. C. Greubel, Aug 05 2022

Formula

From Colin Barker, Dec 05 2016: (Start)

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) for n>5.

G.f.: (1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6.

(End)

E.g.f.: (1114 +2299*x +1382*x^2 +324*x^3 +31*x^4 +x^5)*exp(x). - G. C. Greubel, Aug 05 2022