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%I #31 Sep 24 2022 05:47:48
%S 1,14,168,2016,25200,332640,4656960,69189120,1089728640,18162144000,
%T 319653734400,5928123801600,115598414131200,2365321396838400,
%U 50685458503680000,1135354270482432000,26538906072526848000,646300418472124416000,16372943934627151872000
%N Second (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).
%H Indranil Ghosh, <a href="/A062148/b062148.txt">Table of n, a(n) for n = 0..400</a>
%H Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F E.g.f.: (1+6*x)/(1-x)^8.
%F a(n) = A062138(n+1, 1) = (n+1)!*binomial(n+6, 6).
%F If we define f(n,i,x)= Sum_{k=i..n}(Sum_{j=i..k}(binomial(k,j) *Stirling1(n,k)* Stirling2(j,i)*x^(k-j))) then a(n-1) = (-1)^(n-1) * f(n,1,-7), (n>=1). - _Milan Janjic_, Mar 01 2009
%F Assuming offset 1: a(n) = n!*binomial(-n,6). - _Peter Luschny_, Apr 29 2016
%F From _Amiram Eldar_, Sep 24 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 5477/10 - 204*e - 6*gamma + 6*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
%F Sum_{n>=0} (-1)^n/a(n) = 403/10 - 120/e + 6*gamma - 6*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)
%e a(3) = (3+1)! * binomial(3+6,6) = 24 * 84 = 2016. - _Indranil Ghosh_, Feb 24 2017
%t Table[Sum[n!/6!, {i, 6, n}], {n, 6, 21}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%o (PARI) a(n)=(n+1)!*binomial(n+6,6) \\ _Indranil Ghosh_, Feb 24 2017
%o (Python)
%o import math
%o f=math.factorial
%o def C(n,r):return f(n)/f(r)/f(n-r)
%o def A062148(n): return f(n+1)*C(n+6,6) # _Indranil Ghosh_, Feb 24 2017
%o (Magma) [Factorial(n+1)*Binomial(n+6,6): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y Cf. A001725 (first column of A062138).
%Y Appears in the third left hand column of A167556 multiplied by 120. - _Johannes W. Meijer_, Nov 12 2009
%Y Cf. A001620, A091725, A099285.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jun 19 2001
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