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

1, 1, 2, 18, 244, 5710, 188908, 8702820, 524888040, 40393084950, 3853034107900, 446671026849916, 61824801560228056, 10072685383683311116, 1907978676359896992824, 415795605119514578204616, 103294156408291202467520976, 29018125910193347265466916070

Offset

0,3

Comments

Row sums of Eulerian triangle A008292 yield the factorials.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..253

Formula

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

Example

a(1) = 1 = 1;
a(2) = 1 + 1 = 2;
a(3) = 1 + 4^2 + 1 = 18;
a(4) = 1 + 11^2 + 11^2 + 1 = 244;
a(5) = 1 + 26^2 + 66^2 + 26^2 + 1 = 5710;
a(6) = 1 + 57^2 + 302^2 + 302^2 + 57^2 + 1 = 188908.

Maple

a:= n-> add(combinat[eulerian1](n, k)^2, k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Sep 10 2020

Prog

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

Crossrefs

Cf. A008292.

Column k=2 of A335545.

Sequence in context: A052635 A366001 A259270 * A117514 A138437 A265452

Adjacent sequences: A168559 A168560 A168561 * A168563 A168564 A168565

Keyword

nonn

Author

Paul D. Hanna, Nov 29 2009

Status

approved