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

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, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 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

Offset

0,5

Comments

See A201637, and also A008517 (offset 1 for rows and columns).

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.

For similar triangles see also A112007 and A163936.

Links

Seiichi Manyama, Rows n = 0..139, flattened

Andrew Elvey Price, 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.

Wikipedia, Eulerian numbers of the second kind

Formula

T(n, m) = A201637(n, n-m), n >= m >= 0.

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.

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.

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

Example

The triangle T(n, m) begins:
n\m 0      1       2        3        4       5       6     7    8  9 ...
0:  1
1:  0      1
2:  0      2       1
3:  0      6       8        1
4:  0     24      58       22        1
5:  0    120     444      328       52       1
6:  0    720    3708     4400     1452     114       1
7:  0   5040   33984    58140    32120    5610     240     1
8:  0  40320  341136   785304   644020  195800   19950   494    1
9:  0 362880 3733920 11026296 12440064 5765500 1062500 67260 1004  1
...

Maple

T:= (n, k)-> combinat[eulerian2](n, n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jul 26 2017
# Alternative: using the e.g.f:
alias(W = LambertW): len := 10:
egf := (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1):
ser := simplify(subs(W(-exp(-1/t)/t) = (-1/t), series(egf, x, len+1))):
seq(seq(n!*coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..len);  # Peter Luschny, Mar 13 2025

Mathematica

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 *)

Crossrefs

Cf. A201637, A008517, A112007, A163936.

Columns m = 0..5: A000007, A000142, A002538, A002539, A112008, A112485.

Diagonals d = 0..3: A000012, A005803, A004301, A006260.

T(2n,n) gives A290306.

Sequence in context: A129062 A281662 A163936 * A356545 A375835 A187555

Adjacent sequences: A288871 A288872 A288873 * A288875 A288876 A288877

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Jul 20 2017

Status

approved