A211232 - OEIS

%I #22 May 18 2020 19:01:24

%S 1,2,1,4,1,1,7,0,-7,-1,1,12,-12,-56,-12,12,1,1,21,-67,-284,0,284,67,

%T -21,-1,1,38,-273,-1170,753,3408,753,-1170,-273,38,1,1,71,-982,-4241,

%U 8562,29055,0,-29055,-8562,4241,982,-71,-1,1,136,-3314,-13888,66335,199616,-106113,-464880,-106113,199616,66335,-13888,-3314,136,1

%N Irregular triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 2, for n >= 1 (the rows start at k=1).

%H Andrew Howroyd, <a href="/A211232/b211232.txt">Table of n, a(n) for n = 1..1601</a> (rows 1..40)

%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) for n > 1.

%F A047681(n) = Sum_{k>=1} T(2*n, k).

%F (End)

%e Triangle begins

%e   1,  2;

%e   1,  4,    1;

%e   1,  7,    0,    -7,  -1;

%e   1, 12,  -12,   -56, -12,   12,   1;

%e   1, 21,  -67,  -284,   0,  284,  67,   -21,   -1;

%e   1, 38, -273, -1170, 753, 3408, 753, -1170, -273, 38, 1;

%e   ...

%o (PARI)

%o T(n,r=2)={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(7)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, May 18 2020

%Y Row sums of even rows are A047681; row sums of odd rows are zero for n > 1.

%Y Cf. A008292, A211233, A211234, A211235.

%K sign,tabf

%O 1,2

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

%E Terms a(38) and beyond from _Andrew Howroyd_, May 18 2020