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
Examples
... [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]
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.
Crossrefs
Programs
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Maple
A201637 := (n,k) -> combinat[eulerian2](n,k): for n from 0 to 9 do seq(A201637(n,k),k=0..n) od; # Illustrating the connection with the Lambert W function: alias(W = LambertW): len := 9: w := W(-exp((x - 1)^2 * t - x)*x) + 1: ser := series((1 - 1/x)*(1 - 1/w), t, len + 1): egf := simplify(subs(W(-exp(-x)*x)=(-x), ser)): poly := n -> n!*coeff(egf, t, n): seq(seq(coeff(poly(n), x, k), k = 0..n), n = 0..len); # Peter Luschny, Mar 15 2025
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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 *) -
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
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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)]
Formula
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
Extensions
Terms a(52) onward added by G. C. Greubel, Oct 24 2017
Comments