This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340556 #81 Apr 10 2025 03:07:22
%S A340556 1,0,1,0,1,2,0,1,8,6,0,1,22,58,24,0,1,52,328,444,120,0,1,114,1452,
%T A340556 4400,3708,720,0,1,240,5610,32120,58140,33984,5040,0,1,494,19950,
%U A340556 195800,644020,785304,341136,40320,0,1,1004,67260,1062500,5765500,12440064,11026296,3733920,362880
%N A340556 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.
%C A340556 The second-order Eulerian number E2(n, k) is the number of Stirling permutations of order n with exactly k descents; here the last index is defined to be a descent. More formally, let Q_n denote the set of permutations of the multiset {1,1,2,2, ..., n,n} in which, for all j, all entries between two occurrences of j are larger than j, then E2(n, k) = card({s in Q_n with des(s) = k}), where des(s) = card({j: s(j) > s(j+1)}) is the number of descents of s.
%C A340556 Also the number of Riordan trapezoidal words of length n with k distinct letters (see Riordan 1976, p. 9).
%C A340556 Also the number of rooted plane trees on n + 1 vertices with k leaves (see Janson 2008, p. 543).
%C A340556 Let b(n) = (1/2)*Sum_{k=0..n-1} (-1)^k*E2(n-1, k+1) / C(2*n-1, k+1). Apparently b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1}F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See Rzadkowski and Urlinska, example 4.)
%D A340556 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
%H A340556 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="https://doi.org/10.37236/4814">Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers</a>, Electronic Journal of Combinatorics 22(3), 2015.
%H A340556 Kenny Barrese, Jennifer Elder, Pamela E. Harris, and Anthony Simpson, <a href="https://arxiv.org/abs/2504.06466">Enumerating Flat Fubini Rankings</a>, arXiv:2504.06466 [math.CO], 2025. Conjecture 4.4. p. 14.
%H A340556 Brian Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.10.1)</a>, Thesis, Brandeis Univ., Aug. 2008, p. 58.
%H A340556 Sergi Elizalde, <a href="https://arxiv.org/abs/2002.00985">Descents on quasi-Stirling permutations</a>, arXiv:2002.00985 [math.CO], 2020.
%H A340556 I. Gessel and R. P. Stanley, <a href="https://doi.org/10.1016/0097-3165(78)90042-0">Stirling polynomials</a>, J. Combin. Theory, A 24, 24-33, 1978.
%H A340556 Svante Janson, <a href="https://hal.inria.fr/hal-01194667">Plane recursive trees, Stirling permutations and an urn model</a>, Fifth Colloquium on Mathematics and Computer Science, 541-547, Discrete Math. Theor. Comput. Sci. Proc., AI, 2008.
%H A340556 Peter Luschny, <a href="http://luschny.de/math/oeis/A340556.html">A companion to A340556</a>, A SageMath-Jupyter notebook, Feb. 2021.
%H A340556 Peter Luschny, <a href="https://mathoverflow.net/q/384146">How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?</a>, MathOverflow, Feb. 2021.
%H A340556 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 A340556 John Riordan, <a href="https://doi.org/10.1007/BF02392410">The blossoming of Schröder's fourth problem</a>, Acta Math., 137 (1976), 1-16.
%H A340556 G. Rzadkowski and M. Urlinska, <a href="http://arxiv.org/abs/1612.06635">A Generalization of the Eulerian Numbers</a>, arXiv:1612.06635 [math.CO], 2016.
%F A340556 E2(n, k) = E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) for n > 0 and 0 <= k <= n, and E2(0, 0) = 1; in all other cases E(n, k) = 0.
%F A340556 E2(n, k) = Sum_{j=0..n-k}(-1)^(n-j)*binomial(2*n+1, j)*Stirling1(2*n-k-j+1, n-k-j+1).
%F A340556 E2(n, k) = Sum_{j=0..k}(-1)^(k-j)*binomial(2*n + 1, k - j)*Stirling2(n + j, j).
%F A340556 Stirling1(x, x - n) = (-1)^n*Sum_{k=0..n} E2(n, k)*binomial(x + k - 1, 2*n).
%F A340556 Stirling2(x, x - n) = Sum_{k=0..n} E2(n, k)*binomial(x + n - k, 2*n).
%F A340556 E2poly(n, x) = Sum_{k=0..n} E2(n, k)*x^k, as row polynomials.
%F A340556 E2poly(n, x) = x*(x-1)^(2*n)*d_{x}((1-x)^(1-2*n)*E2poly(n-1)) for n>=1 and E2poly(0)=1.
%F A340556 E2poly(n, x) = (1 - x)^(2*n + 1)*Sum_{k=0..n}(k^(n + k)/k!)*(x*exp(-x))^k.
%F A340556 W(n, k) = [x^k] (1+x)^n*E2poly(n, x/(1 + x)) are the Ward numbers A269939.
%F A340556 E2(n, k) = [x^k] (1-x)^n*Wpoly(n, x/(1 - x)); Wpoly(n, x) = Sum_{k=0..n}W(n, k)*x^k.
%F A340556 W(n, k) = Sum_{j=0..k} E2(n, j)*binomial(n - j, n - k).
%F A340556 E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*W(n, j)*binomial(n - j, k - j).
%F A340556 The compositional inverse with respect to x of x - t*(exp(x) - 1) (see B. Drake):
%F A340556 T(n, k) = [t^k](n+1)!*(1-t)^(2*n+1)*[x^(n+1)] InverseSeries(x - t*(exp(x)-1), x).
%F A340556 AS1(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j+1), where AS1(n, k) are the associated Stirling numbers of the first kind (A008306, A106828).
%F A340556 E2(n, k) = Sum_{j=0..n-k+1} (-1)^(n-k-j+1)*AS1(n+j, j)*binomial(n-j, n-k-j+1), for n >= 1.
%F A340556 AS2(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, n-k-j) for n >=1, where AS2(n, k) are the associated Stirling numbers of the second kind (A008299, A137375).
%F A340556 E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*AS2(n + j, j)*binomial(n - j, k - j).
%e A340556 Triangle starts:
%e A340556 [0] 1;
%e A340556 [1] 0, 1;
%e A340556 [2] 0, 1, 2;
%e A340556 [3] 0, 1, 8, 6;
%e A340556 [4] 0, 1, 22, 58, 24;
%e A340556 [5] 0, 1, 52, 328, 444, 120;
%e A340556 [6] 0, 1, 114, 1452, 4400, 3708, 720;
%e A340556 [7] 0, 1, 240, 5610, 32120, 58140, 33984, 5040;
%e A340556 [8] 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320;
%e A340556 [9] 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880.
%e A340556 To illustrate the generating function for row 3: The expansion of (1 - x)^7*(x*exp(-x) + 16*x^2*exp(-x)^2 + (243*x^3*exp(-x)^3)/2) gives the polynomial x + 8*x^2 + 6*x^3. The coefficients of this polynomial give row 3.
%e A340556 .
%e A340556 Stirling permutations of order 3 with exactly k descents: (When counting the descents one may assume an invisible '0' appended to the permutations.)
%e A340556 T[3, k=0]:
%e A340556 T[3, k=1]: 112233;
%e A340556 T[3, k=2]: 331122; 223311; 221133; 133122; 122331; 122133; 113322; 112332;
%e A340556 T[3, k=3]: 332211; 331221; 233211; 221331; 133221; 123321.
%p A340556 # Using the recurrence:
%p A340556 E2 := proc(n, k) option remember;
%p A340556 if k = 0 and n = 0 then return 1 fi; if n < 0 then return 0 fi;
%p A340556 E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) end: seq(seq(E2(n, k), k = 0..n), n = 0..9);
%p A340556 # Using the row generating function:
%p A340556 E2egf := n -> (1-x)^(2*n+1)*add(k^(n+k)/k!*(x*exp(-x))^k, k=0..n);
%p A340556 T := (n, k) -> coeftayl(E2egf(n), x=0, k): seq(print(seq(T(n, j),j=0..n)), n=0..7);
%p A340556 # Using the built-in function:
%p A340556 E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
%p A340556 # Using the compositional inverse (series reversion):
%p A340556 E2triangle := proc(N) local r, s, C; Order := N + 2;
%p A340556 s := solve(y = series(x - t*(exp(x) - 1), x), x):
%p A340556 r := n -> -n!*(t - 1)^(2*n - 1)*coeff(s, y, n); C := [seq(expand(r(n)), n = 1..N)];
%p A340556 seq(print(seq(coeff(C[n+1], t, k), k = 0..n)), n = 0..N-1) end: E2triangle(10);
%t A340556 T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], If[n < 0, 0, k T[n - 1, k] + (2 n - k) T[n - 1, k - 1]]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%t A340556 (* Via row polynomials: *)
%t A340556 E2poly[n_] := If[n == 0, 1,
%t A340556 Expand[Simplify[x (x - 1)^(2 n) D[((1 - x)^(1 - 2 n) E2poly[n - 1]), x]]]];
%t A340556 Table[CoefficientList[E2poly[n], x], {n, 0, 9}] // Flatten
%t A340556 (* Series reversion *)
%t A340556 Revert[gf_, len_] := Module[{S = InverseSeries[Series[gf, {x, 0, len + 1}], x]},
%t A340556 Table[CoefficientList[(n + 1)! (1 - t)^(2 n + 1) Coefficient[S, x, n + 1], t],
%t A340556 {n, 0, len}] // Flatten]; Revert[x + t - t Exp[x], 6]
%o A340556 (PARI)
%o A340556 E2poly(n) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1)));
%o A340556 { for(n = 0, 9, print(Vecrev(E2poly(n)))) }
%o A340556 (PARI) T(n, k) = sum(j=0, n-k, (-1)^(n-j)*binomial(2*n+1, j)*stirling(2*n-k-j+1, n-k-j+1, 1)); \\ _Michel Marcus_, Feb 11 2021
%o A340556 (SageMath) # See also link to notebook.
%o A340556 @cached_function
%o A340556 def E2(n, k):
%o A340556 if n < 0: return 0
%o A340556 if k == 0: return k^n
%o A340556 return k * E2(n - 1, k) + (2*n - k) * E2(n - 1, k - 1) # _Peter Luschny_, Mar 08 2025
%Y A340556 Indexing the second-order Eulerian numbers comes in three flavors: A008517 (following Riordan and Comtet), A201637 (following Graham, Knuth, and Patashnik) and this indexing, extending the definition of Gessel and Stanley. (A008517 is the main entry of the numbers.) The corresponding triangles of the first-order Eulerian numbers are A008292, A173018, and A123125.
%Y A340556 Row reversed: A163936 (with offset = 0).
%Y A340556 Values: E2poly(n, 1) = A001147(n), E2poly(n, -1) ~ -A001662(n+1), E2poly(n, 2) = A112487(n), 2^n*E2poly(n, 1/2) = A000311(n+1), 2^n*E2poly(n, -1/2) = A341106(n).
%Y A340556 Diagonals: A000142, A002538, A002539, A112008, A112485.
%Y A340556 Columns: A000007, A000012, A005803, A004301, A006260.
%Y A340556 Cf. A008299, A137375, A008306, A106828, A269939, A278075, A331998.
%K A340556 nonn,tabl
%O A340556 0,6
%A A340556 _Peter Luschny_, Feb 05 2021