|
%I #36 Aug 14 2022 05:20:41
%S 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,
%T 3708,720,0,1,240,5610,32120,58140,33984,5040,0,1,494,19950,195800,
%U 644020,785304,341136,40320,0,1,1004,67260,1062500,5765500,12440064,11026296,3733920,362880,0
%N Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
%C 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.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 256.
%H G. C. Greubel, <a href="/A201637/b201637.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H Dengji Qi, <a href="https://ajc.maths.uq.edu.au/pdf/50/ajc_v50_p183.pdf">Note: On the second order Eulerian numbers</a>, Australasian Journal of Combinatorics, Volume 50 (2011), Pages 183-185.
%e ... [0] [1] [2] [3] [4] [5] [6] [7] [8]
%e [0] [1]
%e [1] [1, 0]
%e [2] [1, 2, 0]
%e [3] [1, 8, 6, 0]
%e [4] [1, 22, 58, 24, 0]
%e [5] [1, 52, 328, 444, 120, 0]
%e [6] [1, 114, 1452, 4400, 3708, 720, 0]
%e [7] [1, 240, 5610, 32120, 58140, 33984, 5040, 0]
%e [8] [1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0]
%p A201637 := (n,k) -> combinat[eulerian2](n,k):
%p for n from 0 to 9 do seq(A201637(n,k),k=0..n) od;
%t 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
%t (* _Jean-François Alcover_, Jun 28 2013 *)
%t E2[n_, k_] /; k == 0 = 1; E2[n_, k_] /; k < 0 || k > n = 0;
%t E2[n_, k_] := E2[n, k] = (2*n - 1 - k)*E2[n-1, k-1] + (k + 1)*E2[n-1, k];
%t Table[E2[n, k], {n, 0, 8}, {k, 0, n}] // TableForm
%t (* _Peter Luschny_, Aug 14 2022 *)
%o (Sage)
%o @CachedFunction
%o def eulerian2(n, k):
%o if k==0: return 1
%o if k==n: return 0
%o return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
%o for n in (0..9): [eulerian2(n, k) for k in(0..n)]
%o (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
%Y Cf. A008517, A173018.
%Y Columns 2 and 3 respectively give A004301 and A006260.
%Y T(2n,n) gives A290306.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Nov 11 2012
%E Terms a(52) onward added by _G. C. Greubel_, Oct 24 2017
|
