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A211234 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 4, n >= 1.

%I A211234 #29 May 18 2020 19:02:18
%S A211234 1,2,3,4,1,4,10,20,10,4,1,1,7,27,77,57,0,-57,-77,-27,-7,-1,1,12,69,
%T A211234 272,221,-272,-1084,-1688,-1084,-272,221,272,69,12,1,1,21,176,936,625,
%U A211234 -3288,-11868,-21023,-16223,0,16223,21023,11868,3288,-625,-936,-176,-21,-1
%N A211234 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 4, n >= 1.
%H A211234 Andrew Howroyd, <a href="/A211234/b211234.txt">Table of n, a(n) for n = 1..1276</a> (rows 1..25)
%H A211234 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 A211234 A047683(n) = Sum_{k>=1} T(2*n, k). - _Andrew Howroyd_, May 18 2020
%e A211234 Triangle begins:
%e A211234   1, 2,  3,  4;
%e A211234   1, 4, 10, 20, 10, 4,   1;
%e A211234   1, 7, 27, 77, 57, 0, -57, -77, -27, -7, -1;
%e A211234   ...
%o A211234 (PARI)
%o A211234 T(n,r=4)={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 A211234 { my(A=T(5)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, May 18 2020
%Y A211234 Row sums of even rows are A047683; row sums of odd rows are zero for n > 1.
%Y A211234 Cf. A008292, A211232, A211233, A211235.
%K A211234 sign,tabf
%O A211234 1,2
%A A211234 _N. J. A. Sloane_, Apr 05 2012
%E A211234 More terms from _Franck Maminirina Ramaharo_, Nov 30 2018
%E A211234 a(20) corrected by _Andrew Howroyd_, May 18 2020