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A345651 Fourth column of A008296.

%I A345651 #39 Sep 15 2021 10:12:59
%S A345651 1,10,25,-35,49,0,-820,9020,-87164,859144,-8965320,100136400,
%T A345651 -1199838576,15406135488,-211479420096,3094582896000,-48129022468224,
%U A345651 793274283938304,-13818265424460288,253731538514893824,-4899371564756837376,99261476593521868800
%N A345651 Fourth column of A008296.
%H A345651 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,4).
%F A345651 a(n) = A008296(n,4).
%F A345651 a(n) = (-1)^n*(4*H(n-5,1)^3 + 8*H(n-5,3) - 12*H(n-5,2)*H(n-5,1) - 25*H(n-5,1)^2 + 25*H(n-5,2) + 35*H(n-5,1) - 10)*(n-5)! for n >= 5 where H(n,1) = Sum_{j=1..n} 1/j is the n-th harmonic number, H(n,2) = Sum_{j=1..n} 1/j^2 and H(n,3) = Sum_{j=1..n} 1/j^3.
%F A345651 a(n) = Sum_{m=4..n} binomial(m,4) * 4^(m-4) * Stirling1(n,m). - _Alois P. Heinz_, Aug 26 2021
%F A345651 Conjecture: D-finite with recurrence a(n) +2*(2*n-13)*a(n-1) +(6*n^2-84*n+295)*a(n-2) +(2*n-15)*(2*n^2-30*n+113)*a(n-3) +(n-8)^4*a(n-4)=0. - _R. J. Mathar_, Sep 15 2021
%p A345651 b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
%p A345651       (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
%p A345651     end:
%p A345651 a:= n-> b(n, 4):
%p A345651 seq(a(n), n=4..28);  # _Alois P. Heinz_, Aug 26 2021
%p A345651 # alternative
%p A345651 seq(A008296(n,4),n=4..70) ; # _R. J. Mathar_, Sep 15 2021
%t A345651 a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
%t A345651 a[n_, n_] = 1;
%t A345651 a[n_, k_] := a[n, k] = (n - 1) a[n - 2, k - 1] +
%t A345651     a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
%t A345651 Flatten[Table[N[a[n + 4, 4], 10], {n, 1, 400}]]
%o A345651 (PARI) a(n) = sum(m=4, n, binomial(m, 4)*4^(m-4)*stirling(n, m, 1)); \\ _Michel Marcus_, Sep 14 2021
%Y A345651 Cf. A008296, A045406, A081048.
%Y A345651 Cf. A001008, A002805, A007406, A007407.
%K A345651 sign
%O A345651 4,2
%A A345651 _Luca Onnis_, Aug 26 2021