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A288874
Row reversed version of triangle A201637 (second-order Eulerian triangle).
3
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.
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.
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
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Jul 20 2017
STATUS
approved