A343804 - OEIS

%I #10 Apr 30 2021 10:56:02

%S 1,1,1,1,4,1,1,15,11,1,1,64,96,26,1,1,325,824,448,57,1,1,1956,7417,

%T 6718,1779,120,1,1,13699,71595,96633,43411,6429,247,1,1,109600,746232,

%U 1393588,944618,243928,21898,502,1,1,986409,8403000,20600856,19521210,7739362,1250774,71742,1013,1

%N T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

%e Triangle starts:

%e [0] 1

%e [1] 1, 1

%e [2] 1, 4,      1

%e [3] 1, 15,     11,      1

%e [4] 1, 64,     96,      26,       1

%e [5] 1, 325,    824,     448,      57,       1

%e [6] 1, 1956,   7417,    6718,     1779,     120,     1

%e [7] 1, 13699,  71595,   96633,    43411,    6429,    247,     1

%e [8] 1, 109600, 746232,  1393588,  944618,   243928,  21898,   502,   1

%e [9] 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

%p T := (n, k) -> add(binomial(n, r)*combinat:-eulerian2(r, r-k), r = k..n):

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%Y Row sums: A084262.

%Y Cf. A046802 (Eulerian first order).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Apr 30 2021