A071245 a(n) = n*(n-1)*(2*n^2 + 1)/6.

0, 0, 3, 19, 66, 170, 365, 693, 1204, 1956, 3015, 4455, 6358, 8814, 11921, 15785, 20520, 26248, 33099, 41211, 50730, 61810, 74613, 89309, 106076, 125100, 146575, 170703, 197694, 227766, 261145, 298065, 338768, 383504, 432531, 486115, 544530, 608058

Offset: 0

Author

N. J. A. Sloane, Jun 12 2002

Keywords

• nonn
• easy

Comments

The first differences are given in A277228. - J. M. Bergot, Sep 14 2016

References

• T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A071238, A071244, A277228 (first differences).

Programs

Magma

[n*(n-1)*(2*n^2+1)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Mathematica

Table[n (n - 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)
CoefficientList[Series[(-3 x^2 - 4 x^3 - x^4)/(-1 + x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Sep 14 2016 *)

PARI

a(n)=n*(n-1)*(2*n^2+1)/6; \\ Joerg Arndt, Sep 04 2013

SageMath

def A071245(n): return binomial(n,2)*(2*n^2+1)//3
[A071245(n) for n in range(41)] # G. C. Greubel, Aug 07 2024

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4; a(0)=0, a(1)=0, a(2)=3, a(3)=19, a(4)=66. - Yosu Yurramendi, Sep 03 2013

G.f.: x^2*(3 + 4*x + x^2)/(1-x)^5. - Michael De Vlieger, Sep 14 2016

E.g.f.: (1/6)*x^2*(9 + 10*x + 2*x^2)*exp(x). - G. C. Greubel, Sep 23 2016