A099762 a(n) = n^2 * (n+1)^3.

0, 8, 108, 576, 2000, 5400, 12348, 25088, 46656, 81000, 133100, 209088, 316368, 463736, 661500, 921600, 1257728, 1685448, 2222316, 2888000, 3704400, 4695768, 5888828, 7312896, 9000000, 10985000, 13305708, 16003008, 19120976, 22707000

Offset

0,2

Comments

a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n+1} such that for fixed different x_1, x_2 in {1,2,3,4,5} and fixed y_1, y_2 in {1,2,...,n+1} we have f(x_1)<>y_1 and f(x_2)<>y_2. - Milan Janjic, Apr 17 2007

Pierce expansion of the constant 1 - Sum {k >= 1} (-1)^(k+1)*k^2/k!^5 = 0.12384 46009 75944 78422 ... = 1/8 - 1/(8*108) + 1/(8*108*576) - .... - Peter Bala, Feb 01 2015

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Eric Weisstein's World of Mathematics, Pierce Expansion

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

G.f.: 4*x*(2 +15*x +12*x^2 +x^3)/(1-x)^6. - Colin Barker, Feb 01 2015

E.g.f.: x*(8 +46*x +46*x^2 +13*x^3 +x^4)*exp(x). - G. C. Greubel, Sep 03 2019

From Amiram Eldar, Jul 19 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(3) + Pi^2/2 - 6.

Sum_{n>=1} (-1)^(n+1)/a(n) = 6 - 3*zeta(3)/4 - Pi^2/12 - 6*log(2). (End)

Maple

a:=n->sum(sum(n^3, j=2..n), k=2..n): seq(a(n), n=1..30); # Zerinvary Lajos, May 09 2007

Mathematica

Table[n^2 (n+1)^3, {n, 0, 30}]  (* Harvey P. Dale, Feb 08 2011 *)

Prog

(PARI) vector(30, n, n--; n^2*(n+1)^3) \\ Colin Barker, Feb 01 2015
(Magma) [n^2*(n+1)^3: n in [0..30]]; // G. C. Greubel, Sep 03 2019
(Sage) [n^2*(n+1)^3 for n in (0..30)] # G. C. Greubel, Sep 03 2019
(GAP) List([0..30], n-> n^2*(n+1)^3); # G. C. Greubel, Sep 03 2019

Crossrefs

Sequence in context: A187288 A275134 A187190 * A246498 A119936 A276296

Adjacent sequences: A099759 A099760 A099761 * A099763 A099764 A099765

Keyword

easy,nonn

Author

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

Status

approved