A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

1, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880, 0

Offset

0,5

Comments

This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics. This indexing is also used by Maple. The indexing as used by Riordan, Comtet and others, is given in A008517, which is the main entry for the second-order Eulerian numbers.

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 256.

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

Dengji Qi, Note: On the second order Eulerian numbers, Australasian Journal of Combinatorics, Volume 50 (2011), Pages 183-185.

Example

... [0]  [1]    [2]     [3]     [4]     [5]     [6]    [7]  [8]
[0] [1]
[1] [1,   0]
[2] [1,   2,     0]
[3] [1,   8,     6,      0]
[4] [1,  22,    58,     24,      0]
[5] [1,  52,   328,    444,    120,      0]
[6] [1, 114,  1452,   4400,   3708,    720,      0]
[7] [1, 240,  5610,  32120,  58140,  33984,   5040,     0]
[8] [1, 494, 19950, 195800, 644020, 785304, 341136, 40320,  0]

Maple

A201637 := (n, k) -> combinat[eulerian2](n, k):
for n from 0 to 9 do seq(A201637(n, k), k=0..n) od;

Mathematica

t[0, 0] = 1; t[n_, m_] = Sum[(-1)^(n+k)*Binomial[2*n+1, k]*StirlingS1[2*n-m-k, n-m-k], {k, 0, n-m-1}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten
(* Jean-François Alcover, Jun 28 2013 *)
E2[n_, k_] /; k == 0 = 1; E2[n_, k_] /; k < 0 || k > n = 0;
E2[n_, k_] := E2[n, k] = (2*n - 1 - k)*E2[n-1, k-1] + (k + 1)*E2[n-1, k];
Table[E2[n, k], {n, 0, 8}, {k, 0, n}] // TableForm
(* Peter Luschny, Aug 14 2022 *)

Prog

(Sage)
@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    if k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
for n in (0..9): [eulerian2(n, k) for k in(0..n)]
(PARI) for(n=0, 10, for(m=0, n, print1(if(m==0 || n==0, 1, sum(k=0, n-m-1, (-1)^(n+k)* binomial(2*n+1, k)*stirling(2*n-m-k, n-m-k, 1))), ", "))) \\ G. C. Greubel, Oct 24 2017

Crossrefs

Cf. A008517, A173018.

Columns 2 and 3 respectively give A004301 and A006260.

T(2n,n) gives A290306.

Sequence in context: A344069 A337444 A340556 * A055141 A055140 A335330

Adjacent sequences: A201634 A201635 A201636 * A201638 A201639 A201640

Keyword

nonn,tabl

Author

Peter Luschny, Nov 11 2012

Extensions

Terms a(52) onward added by G. C. Greubel, Oct 24 2017

Status

approved