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A211235
Array of generalized Eulerian numbers C(n,k) read by antidiagonals.
4
1, 1, 2, 1, 4, 3, 1, 7, 10, 4, 1, 12, 27, 20, 5, 1, 21, 69, 77, 35, 6, 1, 38, 176, 272, 182, 56, 7, 1, 71, 456, 936, 846, 378, 84, 8, 1, 136, 1205, 3210, 3750, 2232, 714, 120, 9, 1, 265, 3247, 11075, 16290, 12342, 5214, 1254, 165, 10
OFFSET
1,3
LINKS
D. H. Lehmer, Generalized Eulerian numbers, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).
FORMULA
From Peter Bala, Oct 27 2015: (Start)
O.g.f. of n-th row of square array: 1/(1 - x)^n * (x*d/dx)^n (log(1/(1 - x)), for n >= 1.
E.g.f. of square array: log((1 - x)/(1 - x*exp(t/(1 - x)))).
Read as a triangle: T(n,k) = Sum_{i = 1..k} binomial(n-i,k-i)*i^(n-k) for 1 <= k <= n.
n-th row polynomial of triangle: Sum_{i = 0..n-1} x^i*(x + i)^(n-i). (End)
EXAMPLE
Array begins
1, 2, 3, 4, 5, 6, ... A000027
1, 4, 10, 20, 35, 56, ... A000292
1, 7, 27, 77, 182, 378, ... A005585
1, 12, 69, 272, 846, 2232, ... A101097
1, 21, 176, 936, 3750, 12342, ... A254681
...
Triangle begins
1
1 2
1 4 3
1 7 10 4
1 12 27 20 5
1 21 69 77 35 6
1 38 176 272 182 56 7
...
MAPLE
A211235 := (n, k) -> add(binomial(n-i, k-i)*i^(n-k), i = 1 .. k): for n from 1 to 10 do seq(A211235(n, k), k = 1 .. n) end do; # Peter Bala, Oct 27 2015
MATHEMATICA
T[n_, k_] := Sum[Binomial[n-i, k-i] * i^(n-k), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] //Flatten (* Amiram Eldar, Nov 30 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Apr 05 2012
EXTENSIONS
Terms a(37)-a(55) added by Peter Bala, Oct 27 2015
STATUS
approved