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A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows. 22
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 (list; table; graph; refs; listen; history; text; internal format)
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
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
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
Columns 2 and 3 respectively give A004301 and A006260.
T(2n,n) gives A290306.
Sequence in context: A344069 A337444 A340556 * A055141 A055140 A335330
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 11 2012
EXTENSIONS
Terms a(52) onward added by G. C. Greubel, Oct 24 2017
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)