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A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.

%I A168562 #8 Sep 14 2020 17:03:22
%S A168562 1,1,2,18,244,5710,188908,8702820,524888040,40393084950,3853034107900,
%T A168562 446671026849916,61824801560228056,10072685383683311116,
%U A168562 1907978676359896992824,415795605119514578204616,103294156408291202467520976,29018125910193347265466916070
%N A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.
%C A168562 Row sums of Eulerian triangle A008292 yield the factorials.
%H A168562 Alois P. Heinz, <a href="/A168562/b168562.txt">Table of n, a(n) for n = 0..253</a>
%F A168562 a(n) = Sum_{k=0..n} [ Sum_{j=0..k} (-1)^j*(k-j)^n*C(n+1, j) ]^2.
%e A168562 a(1) = 1 = 1;
%e A168562 a(2) = 1 + 1 = 2;
%e A168562 a(3) = 1 + 4^2 + 1 = 18;
%e A168562 a(4) = 1 + 11^2 + 11^2 + 1 = 244;
%e A168562 a(5) = 1 + 26^2 + 66^2 + 26^2 + 1 = 5710;
%e A168562 a(6) = 1 + 57^2 + 302^2 + 302^2 + 57^2 + 1 = 188908.
%p A168562 a:= n-> add(combinat[eulerian1](n, k)^2, k=0..n):
%p A168562 seq(a(n), n=0..18);  # _Alois P. Heinz_, Sep 10 2020
%o A168562 (PARI) {a(n)=sum(k=0,n,sum(j=0, k, (-1)^j*(k-j)^n*binomial(n+1, j))^2)}
%Y A168562 Cf. A008292.
%Y A168562 Column k=2 of A335545.
%K A168562 nonn
%O A168562 0,3
%A A168562 _Paul D. Hanna_, Nov 29 2009