A172080 a(n) = n*(12*n^3 + 10*n^2 - 9*n - 7)/6.

0, 1, 37, 190, 590, 1415, 2891, 5292, 8940, 14205, 21505, 31306, 44122, 60515, 81095, 106520, 137496, 174777, 219165, 271510, 332710, 403711, 485507, 579140, 685700, 806325, 942201, 1094562, 1264690, 1453915, 1663615, 1895216, 2150192

Offset: 0

Author

Vincenzo Librandi, Jan 25 2010

Keywords

• nonn
• easy

Comments

The sequence is related to A172078 by a(n) = n*A172078(n) - Sum_{i=0..n-1} A172078(i).

This is the case d=8 in the identity n^2*(n+1)*(2*d*n-2*d+3)/6 - Sum_{k=0..n-1} k*(k+1)*(2*d*k - 2*d + 3)/6 = n*(n+1)*(3*d*n^2 - d*n + 4*n - 2*d + 2)/12. - Bruno Berselli, May 07 2010, Feb 26 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A172078.

Programs

GAP

List([0..40], n-> n*(n+1)*(12*n^2 -2*n -7)/6); G. C. Greubel, Aug 30 2019

Magma

[(12*n^4+10*n^3-9*n^2-7*n)/6: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

Maple

seq(n*(n+1)*(12*n^2 -2*n -7)/6, n=0..40); # G. C. Greubel, Aug 30 2019

Mathematica

CoefficientList[Series[x(1+32x+15x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)
Table[n*(n+1)*(12*n^2 -2*n -7)/6, {n,0,40}] (* G. C. Greubel, Aug 30 2019 *)

PARI

vector(40, n, n*(n-1)*(12*(n-1)^2 -2*n -5)/6) \\ G. C. Greubel, Aug 30 2019

Sage

[n*(n+1)*(12*n^2 -2*n -7)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

a(n) = n*(n+1)*(12*n^2 - 2*n - 7)/6.

G.f.: x*(1 + 32*x + 15*x^2)/(1-x)^5. - Bruno Berselli, Feb 26 2011

E.g.f.: x*(6 + 105*x + 82*x^2 + 12*x^3)*exp(x)/6. - G. C. Greubel, Aug 30 2019