cp's OEIS Frontend

A347276 Third column of A008296.

%I A347276 #44 Jun 20 2025 16:32:53
%S A347276 1,6,5,-15,49,-196,944,-5340,34716,-254760,2078856,-18620784,
%T A347276 180973584,-1887504768,20887922304,-242111586816,2889841121280,
%U A347276 -34586897978880,393722260047360,-3659128846433280,5687630494110720,1137542166526464000,-49644151627682304000
%N A347276 Third column of A008296.
%H A347276 Louis Comtet, <a href="https://archive.org/details/Comtet_Louis_-_Advanced_Coatorics/page/n73/mode/2up">Advanced Combinatorics</a>, Reidel, 1974, p. 139, b(n,3).
%F A347276 a(n) = A008296(n,3).
%F A347276 a(n) = (-1)^n*(3*H(n-4,1)^2 - 3*H(n-4,2) - 11*H(n-4,1) + 6)*(n-4)! for n >= 4, where H(n,1) = Sum_{j=1..n} 1/j = A001008(n)/A002805(n) is the n-th harmonic number and H(n,2) = Sum_{j=1..n} 1/j^2 = A007406(n)/A007407(n).
%F A347276 a(n) = Sum_{m=3..n} binomial(m,3) * 3^(m-3) * Stirling1(n,m). - _Alois P. Heinz_, Aug 26 2021
%p A347276 b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
%p A347276       (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
%p A347276     end:
%p A347276 a:= n-> b(n, 3):
%p A347276 seq(a(n), n=3..30);  # _Alois P. Heinz_, Aug 25 2021
%t A347276 a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
%t A347276 a[n_, n_] = 1;
%t A347276 a[n_, k_] :=  a[n, k] = (n - 1) a[n - 2, k - 1] +
%t A347276     a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
%t A347276 Flatten[Table[a[n + 3, 3], {n, 0, 30}]]
%o A347276 (PARI) a(n) = sum(m=3, n, binomial(m, 3)*3^(m-3)*stirling(n, m, 1)); \\ _Michel Marcus_, Sep 14 2021
%Y A347276 Cf. A008296, A045406, A081048.
%Y A347276 Cf. A001008, A002805, A007406, A007407.
%Y A347276 Cf. A048994, A008275.
%K A347276 sign
%O A347276 3,2
%A A347276 _Luca Onnis_, Aug 25 2021