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

%I A201637 #38 Mar 15 2025 06:11:15
%S A201637 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 A201637 3708,720,0,1,240,5610,32120,58140,33984,5040,0,1,494,19950,195800,
%U A201637 644020,785304,341136,40320,0,1,1004,67260,1062500,5765500,12440064,11026296,3733920,362880,0
%N A201637 Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
%C A201637 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 A201637 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 256.
%H A201637 G. C. Greubel, <a href="/A201637/b201637.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H A201637 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 A201637 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 A201637 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.
%F A201637 T(n, k) = [x^k](n! * [t^n](1 - 1/x)*(1 - 1/w)), where w = W(-exp((x - 1)^2 * t - x)*x) + 1, and W(-exp(-x)*x) is substituted after expansion by (-x). (W is the Lambert W function.) - _Peter Luschny_, Mar 15 2025
%e A201637 ... [0]  [1]    [2]     [3]     [4]     [5]     [6]    [7]  [8]
%e A201637 [0] [1]
%e A201637 [1] [1,   0]
%e A201637 [2] [1,   2,     0]
%e A201637 [3] [1,   8,     6,      0]
%e A201637 [4] [1,  22,    58,     24,      0]
%e A201637 [5] [1,  52,   328,    444,    120,      0]
%e A201637 [6] [1, 114,  1452,   4400,   3708,    720,      0]
%e A201637 [7] [1, 240,  5610,  32120,  58140,  33984,   5040,     0]
%e A201637 [8] [1, 494, 19950, 195800, 644020, 785304, 341136, 40320,  0]
%p A201637 A201637 := (n,k) -> combinat[eulerian2](n,k):
%p A201637 for n from 0 to 9 do seq(A201637(n,k),k=0..n) od;
%p A201637 # Illustrating the connection with the Lambert W function:
%p A201637 alias(W = LambertW): len := 9:
%p A201637 w := W(-exp((x - 1)^2 * t - x)*x) + 1:
%p A201637 ser := series((1 - 1/x)*(1 - 1/w), t, len + 1):
%p A201637 egf := simplify(subs(W(-exp(-x)*x)=(-x), ser)):
%p A201637 poly := n -> n!*coeff(egf, t, n):
%p A201637 seq(seq(coeff(poly(n), x, k), k = 0..n), n = 0..len);  # _Peter Luschny_, Mar 15 2025
%t A201637 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 A201637 (* _Jean-François Alcover_, Jun 28 2013 *)
%t A201637 E2[n_, k_] /; k == 0 = 1; E2[n_, k_] /; k < 0 || k > n = 0;
%t A201637 E2[n_, k_] := E2[n, k] = (2*n - 1 - k)*E2[n-1, k-1] + (k + 1)*E2[n-1, k];
%t A201637 Table[E2[n, k], {n, 0, 8}, {k, 0, n}] // TableForm
%t A201637 (* _Peter Luschny_, Aug 14 2022 *)
%o A201637 (Sage)
%o A201637 @CachedFunction
%o A201637 def eulerian2(n, k):
%o A201637     if k==0: return 1
%o A201637     if k==n: return 0
%o A201637     return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
%o A201637 for n in (0..9): [eulerian2(n, k) for k in(0..n)]
%o A201637 (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 A201637 Cf. A008517, A173018.
%Y A201637 Columns 2 and 3 respectively give A004301 and A006260.
%Y A201637 T(2n,n) gives A290306.
%K A201637 nonn,tabl
%O A201637 0,5
%A A201637 _Peter Luschny_, Nov 11 2012
%E A201637 Terms a(52) onward added by _G. C. Greubel_, Oct 24 2017