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A211233 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.

%I A211233 #17 May 18 2020 19:01:46
%S A211233 1,2,3,1,4,10,4,1,1,7,27,13,-13,-27,-7,-1,1,12,69,16,-182,-376,-182,
%T A211233 16,69,12,1,1,21,176,-88,-1375,-3123,-1608,1608,3123,1375,88,-176,-21,
%U A211233 -1,1,38,456,-886,-8292,-20322,-6536,35890,65862,35890,-6536,-20322,-8292,-886,456,38,1
%N A211233 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.
%H A211233 Andrew Howroyd, <a href="/A211233/b211233.txt">Table of n, a(n) for n = 1..1366</a> (rows 1..30)
%H A211233 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 A211233 From _Andrew Howroyd_, May 18 2020: (Start)
%F A211233 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 A211233 A047682(n) = Sum_{k>=1} T(2*n, k).
%F A211233 (End)
%e A211233 Triangle begins
%e A211233   1,  2,   3;
%e A211233   1,  4,  10,   4,     1;
%e A211233   1,  7,  27,  13,   -13,   -27,    -7,   -1;
%e A211233   1, 12,  69,  16,  -182,  -376,  -182,   16,   69,   12,  1;
%e A211233   1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, ... ;
%e A211233   ...
%o A211233 (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 A211233 {my(A=T(5)); for(n=1, #A, print(A[n]))} \\ _Andrew Howroyd_, May 18 2020
%Y A211233 Row sums of even rows are A047682; row sums of odd rows are zero for n > 1.
%Y A211233 Cf. A008292, A211232, A211234, A211235.
%K A211233 sign,tabf
%O A211233 1,2
%A A211233 _N. J. A. Sloane_, Apr 05 2012
%E A211233 Terms a(39) and beyond from _Andrew Howroyd_, May 18 2020