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A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).

%I A288874 #27 Mar 13 2025 16:07:47
%S A288874 1,0,1,0,2,1,0,6,8,1,0,24,58,22,1,0,120,444,328,52,1,0,720,3708,4400,
%T A288874 1452,114,1,0,5040,33984,58140,32120,5610,240,1,0,40320,341136,785304,
%U A288874 644020,195800,19950,494,1,0,362880,3733920,11026296,12440064,5765500,1062500,67260,1004,1,0,3628800,44339040,162186912,238904904,155357384,44765000,5326160,218848,2026,1
%N A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).
%C A288874 See A201637, and also A008517 (offset 1 for rows and columns).
%C A288874 The row polynomials of this triangle P(n, x) = Sum_{m=0..n} T(n, m)*x^m appear as numerator polynomials in the o.g.f.s for the diagonal sequences of triangle A132393 (|Stirling1| with offset 0 for rows and columns). See the comment and the P. Bala link there.
%C A288874 For similar triangles see also A112007 and A163936.
%H A288874 Seiichi Manyama, <a href="/A288874/b288874.txt">Rows n = 0..139, flattened</a>
%H A288874 Andrew Elvey Price, 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 A288874 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eulerian_number#Eulerian_numbers_of_the_second_kind">Eulerian numbers of the second kind</a>
%F A288874 T(n, m) = A201637(n, n-m), n >= m >= 0.
%F A288874 Recurrence: T(0, 0) = 1, T(n, -1) = 0, T(n, m) = 0 if n < m, (n-m+1)*T(n-1, m-1) + (n-1+m)*T(n-1, m), n >= 1, m = 0..n; from the A008517 recurrence.
%F A288874 T(0, 0) = 1, T(n, m) = Sum_{p = 0..m-1} (-1)^(n-p)*binomial(2*n+1, p)*A132393(n+m-p, m-p), n >= 1, m = 0..n; from a A008517 program.
%F A288874 T(n, k) = n! * [t^k][x^n] (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1) where after expansion W(-exp(-1/t)/t) is substituted by (-1/t). [Inspired by a formula of _Shamil Shakirov_ in A008517.] - _Peter Luschny_, Mar 13 2025
%e A288874 The triangle T(n, m) begins:
%e A288874 n\m 0      1       2        3        4       5       6     7    8  9 ...
%e A288874 0:  1
%e A288874 1:  0      1
%e A288874 2:  0      2       1
%e A288874 3:  0      6       8        1
%e A288874 4:  0     24      58       22        1
%e A288874 5:  0    120     444      328       52       1
%e A288874 6:  0    720    3708     4400     1452     114       1
%e A288874 7:  0   5040   33984    58140    32120    5610     240     1
%e A288874 8:  0  40320  341136   785304   644020  195800   19950   494    1
%e A288874 9:  0 362880 3733920 11026296 12440064 5765500 1062500 67260 1004  1
%e A288874 ...
%p A288874 T:= (n, k)-> combinat[eulerian2](n, n-k):
%p A288874 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Jul 26 2017
%p A288874 # Using the e.g.f:
%p A288874 alias(W = LambertW): len := 10:
%p A288874 egf := (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1):
%p A288874 ser := simplify(subs(W(-exp(-1/t)/t) = (-1/t), series(egf, x, len+1))):
%p A288874 seq(seq(n!*coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..len);  # _Peter Luschny_, Mar 13 2025
%t A288874 Table[Boole[n == 0] + Sum[(-1)^(n + k) * Binomial[2 n + 1, k] StirlingS1[2 n - m - k, n - m - k], {k, 0, n - m - 1}], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* _Michael De Vlieger_, Jul 21 2017, after _Jean-François Alcover_ at A201637 *)
%Y A288874 Cf. A201637, A008517, A112007, A163936.
%Y A288874 Columns m = 0..5: A000007, A000142, A002538, A002539, A112008, A112485.
%Y A288874 Diagonals d = 0..3: A000012, A005803, A004301, A006260.
%Y A288874 T(2n,n) gives A290306.
%K A288874 nonn,easy,tabl
%O A288874 0,5
%A A288874 _Wolfdieter Lang_, Jul 20 2017