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a(n) = ((n+3)!/2)*Sum_{k=1..n} (-1)^(k+1)/(k+3).
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%I #20 Feb 11 2024 23:44:54

%S 3,3,78,186,4008,15912,340560,1931760,43139520,321312960,7611891840,

%T 70589232000,1783264896000,19854108288000,535217663232000,

%U 6967948748544000,200181525175296000,2987361024592896000,91267413626898432000,1537150149529860096000,49817611958159130624000

%N a(n) = ((n+3)!/2)*Sum_{k=1..n} (-1)^(k+1)/(k+3).

%H Andrew Howroyd, <a href="/A024189/b024189.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%F a(n) = ((n+3)!/12)*(5 - 6*log(2) + 3*(-1)^n*(psi((n+4)/2) - psi((n+5)/2))), psi(x) is the digamma function. - _G. C. Greubel_, Jan 02 2020

%p seq( (n+3)!*(5 + 6*add((-1)^k/k, k=1..n+3))/12, n=1..25); # _G. C. Greubel_, Jan 02 2020

%t Table[(n+3)!*(5 + 6*Sum[(-1)^k/k, {k, n+3}])/12, {n, 25}] (* _G. C. Greubel_, Jan 02 2020 *)

%o (PARI) a(n) = (n+3)!/2*sum(x=1, n, (-1)^(x+1)/(x+3)) \\ _Michel Marcus_, Mar 21 2013

%o (Magma) [Factorial(n+3)*(5 + 6*(&+[(-1)^k/k: k in [1..n+3]]))/12: n in [1..25]]; // _G. C. Greubel_, Jan 02 2020

%o (Sage) [factorial(n+3)*(5 + 6*sum((-1)^k/k for k in (1..n+3)))/12 for n in (1..25)] # _G. C. Greubel_, Jan 02 2020

%o (GAP) List([1..25], n-> Factorial(n+3)*(5 + 6*Sum([1..n+3], k-> (-1)^k/k))/12 ); # _G. C. Greubel_, Jan 02 2020

%Y Cf. A024188.

%K nonn

%O 1,1

%A _Clark Kimberling_

%E Terms a(14) and beyond from _Andrew Howroyd_, Jan 01 2020