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A361893
Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.
1
1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880
OFFSET
0,5
FORMULA
T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 2, 2;
[3] 0, 3, 12, 6;
[4] 0, 4, 36, 72, 24;
[5] 0, 5, 80, 360, 480, 120;
[6] 0, 6, 150, 1200, 3600, 3600, 720;
[7] 0, 7, 252, 3150, 16800, 37800, 30240, 5040;
[8] 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320;
[9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;
MAPLE
A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n, k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);
CROSSREFS
Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.
Sequence in context: A350266 A376724 A375470 * A244129 A363907 A342987
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 28 2023
STATUS
approved