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%I #29 Nov 30 2019 02:06:36
%S 0,1,7,37,201,1231,8653,69273,623521,6235291,68588301,823059733,
%T 10699776673,149796873591,2246953104061,35951249665201,
%U 611171244308673,11001082397556403,209020565553571981,4180411311071439981,87788637532500240001,1931350025715005280463
%N a(n) is the sum over each permutation of S_n of the least element of the descent set.
%C a(1) = 0 since the descent set of the identity permutation is empty.
%C Lim_{n->infinity} a(n)/n! = e - 1.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2011; see Section 1.4, pp. 38.
%H Peter Kagey, <a href="/A319013/b319013.txt">Table of n, a(n) for n = 1..400</a>
%F a(n) = Sum_{k=1..n-1} k^2*binomial(n, k+1)*(n - k - 1)!.
%F a(n+1) = (n+1)*a(n) + n^2, with a(1) = 0.
%F a(n) = A002627(n) - n.
%e For n = 3, the least element of the descent set for each permutation in S_3 is given by the table:
%e +-------------+-------------+----------------------+
%e | permutation | descent set | least element (or 0) |
%e +-------------+-------------+----------------------+
%e | 123 | {} | 0 |
%e | 132 | {2} | 2 |
%e | 213 | {1} | 1 |
%e | 231 | {2} | 2 |
%e | 312 | {1} | 1 |
%e | 321 | {1,2} | 1 |
%e +-------------+-------------+----------------------+
%e Thus a(3) = 0 + 2 + 1 + 2 + 1 + 1 = 7.
%t Table[Sum[k^2*Binomial[n, k + 1]*(n - k - 1)!, {k, 1, n - 1}], {n, 1, 15}]
%o (PARI) a(n) = sum(k=1, n-1, k^2*binomial(n, k+1)*(n-k-1)!); \\ _Michel Marcus_, Nov 28 2019
%Y Cf. A002627.
%K nonn
%O 1,3
%A _Peter Kagey_, Sep 07 2018
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