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A344398 a(n) = (-1)^n * F_{n}((-1)^n * n), where F_{n}(x) is the Fubini polynomial.

%I A344398 #12 May 09 2024 07:39:21
%S A344398 1,1,10,111,8676,243005,49729758,2634606331,1026912225160,
%T A344398 88276603008249,55954905981282210,7103694104486331671,
%U A344398 6655958151527584785900,1171100778886715057133493,1521436331153097968932487206,354408430829377435361459172915,609729139653483641913607434550800
%N A344398 a(n) = (-1)^n * F_{n}((-1)^n * n), where F_{n}(x) is the Fubini polynomial.
%p A344398 F := proc(n) option remember; if n = 0 then return 1 fi;
%p A344398 expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
%p A344398 a := n -> (-1)^n*subs(x = (-1)^n*n, F(n)):
%p A344398 seq(a(n), n = 0..17);
%t A344398 F[n_][x_] := If[n == 0, 1, Sum[k! StirlingS2[n, k] x^k, {k, 0, n}]];
%t A344398 a[n_] := (-1)^n F[n][(-1)^n*n];
%t A344398 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, May 09 2024 *)
%o A344398 (SageMath)
%o A344398 @cached_function
%o A344398 def F(n):
%o A344398     R.<x> = PolynomialRing(ZZ)
%o A344398     if n == 0: return R(1)
%o A344398     return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
%o A344398 def a(n):
%o A344398     return (-1)^n*F(n).substitute(x = (-1)^n*n)
%o A344398 print([a(n) for n in range(17)])
%Y A344398 The coefficients of the Fubini polynomials are A131689.
%Y A344398 Cf. A094420.
%K A344398 nonn
%O A344398 0,3
%A A344398 _Peter Luschny_, May 21 2021