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

%I #27 Feb 16 2025 08:32:55

%S 0,8,108,576,2000,5400,12348,25088,46656,81000,133100,209088,316368,

%T 463736,661500,921600,1257728,1685448,2222316,2888000,3704400,4695768,

%U 5888828,7312896,9000000,10985000,13305708,16003008,19120976,22707000

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

%C 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

%C 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

%H Colin Barker, <a href="/A099762/b099762.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: 4*x*(2 +15*x +12*x^2 +x^3)/(1-x)^6. - _Colin Barker_, Feb 01 2015

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

%F From _Amiram Eldar_, Jul 19 2020: (Start)

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

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

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

%t Table[n^2 (n+1)^3,{n,0,30}] (* _Harvey P. Dale_, Feb 08 2011 *)

%o (PARI) vector(30, n, n--; n^2*(n+1)^3) \\ _Colin Barker_, Feb 01 2015

%o (Magma) [n^2*(n+1)^3: n in [0..30]]; // _G. C. Greubel_, Sep 03 2019

%o (Sage) [n^2*(n+1)^3 for n in (0..30)] # _G. C. Greubel_, Sep 03 2019

%o (GAP) List([0..30], n-> n^2*(n+1)^3); # _G. C. Greubel_, Sep 03 2019

%K easy,nonn,changed

%O 0,2

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