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Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.
4

%I #17 May 18 2020 19:01:46

%S 1,2,3,1,4,10,4,1,1,7,27,13,-13,-27,-7,-1,1,12,69,16,-182,-376,-182,

%T 16,69,12,1,1,21,176,-88,-1375,-3123,-1608,1608,3123,1375,88,-176,-21,

%U -1,1,38,456,-886,-8292,-20322,-6536,35890,65862,35890,-6536,-20322,-8292,-886,456,38,1

%N Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.

%H Andrew Howroyd, <a href="/A211233/b211233.txt">Table of n, a(n) for n = 1..1366</a> (rows 1..30)

%H D. H. Lehmer, <a href="https://doi.org/10.1016/0097-3165(82)90020-6">Generalized Eulerian numbers</a>, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).

%F From _Andrew Howroyd_, May 18 2020: (Start)

%F T(n,k) = k*T(n-1,k) - (n-k)*T(n-1,k-1) - (2*n-k)*T(n-1,k-2) - (3*n-k)*T(n-1,k-3) for n > 1.

%F A047682(n) = Sum_{k>=1} T(2*n, k).

%F (End)

%e Triangle begins

%e 1, 2, 3;

%e 1, 4, 10, 4, 1;

%e 1, 7, 27, 13, -13, -27, -7, -1;

%e 1, 12, 69, 16, -182, -376, -182, 16, 69, 12, 1;

%e 1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, ... ;

%e ...

%o (PARI) T(n,r=3)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}

%o {my(A=T(5)); for(n=1, #A, print(A[n]))} \\ _Andrew Howroyd_, May 18 2020

%Y Row sums of even rows are A047682; row sums of odd rows are zero for n > 1.

%Y Cf. A008292, A211232, A211234, A211235.

%K sign,tabf

%O 1,2

%A _N. J. A. Sloane_, Apr 05 2012

%E Terms a(39) and beyond from _Andrew Howroyd_, May 18 2020