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

1, 2, 3, 1, 4, 10, 4, 1, 1, 7, 27, 13, -13, -27, -7, -1, 1, 12, 69, 16, -182, -376, -182, 16, 69, 12, 1, 1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, -176, -21, -1, 1, 38, 456, -886, -8292, -20322, -6536, 35890, 65862, 35890, -6536, -20322, -8292, -886, 456, 38, 1

Offset: 1

Author

N. J. A. Sloane, Apr 05 2012

Keywords

• sign
• tabf

Examples

			Triangle begins
  1,  2,   3;
  1,  4,  10,   4,     1;
  1,  7,  27,  13,   -13,   -27,    -7,   -1;
  1, 12,  69,  16,  -182,  -376,  -182,   16,   69,   12,  1;
  1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, ... ;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1366 (rows 1..30)
• 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 A047682; row sums of odd rows are zero for n > 1.

Cf. A008292, A211232, A211234, A211235.

Programs

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}
{my(A=T(5)); 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) - (3*n-k)*T(n-1,k-3) for n > 1.

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

(End)

Extensions

Terms a(39) and beyond from Andrew Howroyd, May 18 2020