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A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1. 4

%I

%S 1,1,2,18,244,5710,188908,8702820,524888040,40393084950,3853034107900,

%T 446671026849916,61824801560228056,10072685383683311116,

%U 1907978676359896992824,415795605119514578204616,103294156408291202467520976,29018125910193347265466916070

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

%C Row sums of Eulerian triangle A008292 yield the factorials.

%H Alois P. Heinz, <a href="/A168562/b168562.txt">Table of n, a(n) for n = 0..253</a>

%F a(n) = Sum_{k=0..n} [ Sum_{j=0..k} (-1)^j*(k-j)^n*C(n+1, j) ]^2.

%e a(1) = 1 = 1;

%e a(2) = 1 + 1 = 2;

%e a(3) = 1 + 4^2 + 1 = 18;

%e a(4) = 1 + 11^2 + 11^2 + 1 = 244;

%e a(5) = 1 + 26^2 + 66^2 + 26^2 + 1 = 5710;

%e a(6) = 1 + 57^2 + 302^2 + 302^2 + 57^2 + 1 = 188908.

%p a:= n-> add(combinat[eulerian1](n, k)^2, k=0..n):

%p seq(a(n), n=0..18); # _Alois P. Heinz_, Sep 10 2020

%o (PARI) {a(n)=sum(k=0,n,sum(j=0, k, (-1)^j*(k-j)^n*binomial(n+1, j))^2)}

%Y Cf. A008292.

%Y Column k=2 of A335545.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 29 2009

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Last modified September 28 06:05 EDT 2021. Contains 347703 sequences. (Running on oeis4.)