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
KEYWORD
easy,nonn
AUTHOR
Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004
STATUS
approved