A211232 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).

1, 2, 1, 4, 1, 1, 7, 0, -7, -1, 1, 12, -12, -56, -12, 12, 1, 1, 21, -67, -284, 0, 284, 67, -21, -1, 1, 38, -273, -1170, 753, 3408, 753, -1170, -273, 38, 1, 1, 71, -982, -4241, 8562, 29055, 0, -29055, -8562, 4241, 982, -71, -1, 1, 136, -3314, -13888, 66335, 199616, -106113, -464880, -106113, 199616, 66335, -13888, -3314, 136, 1

Offset: 1

Author

N. J. A. Sloane, Apr 05 2012

Keywords

• sign
• tabf

Examples

			Triangle begins
  1,  2;
  1,  4,    1;
  1,  7,    0,    -7,  -1;
  1, 12,  -12,   -56, -12,   12,   1;
  1, 21,  -67,  -284,   0,  284,  67,   -21,   -1;
  1, 38, -273, -1170, 753, 3408, 753, -1170, -273, 38, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1601 (rows 1..40)
• D. H. Lehmer, Generalized Eulerian numbers, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).

Crossrefs

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

Cf. A008292, A211233, A211234, A211235.

Programs

PARI

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}
{ my(A=T(7)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 18 2020

Formula

From Andrew Howroyd, May 18 2020: (Start)

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.

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

(End)

Extensions

Terms a(38) and beyond from Andrew Howroyd, May 18 2020