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A362369
Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.
2
1, 0, 0, 2, 0, 3, 0, 4, 12, 0, 5, 60, 0, 6, 210, 120, 0, 7, 630, 1260, 0, 8, 1736, 8400, 1680, 0, 9, 4536, 45360, 30240, 0, 10, 11430, 216720, 327600, 30240, 0, 11, 28050, 956340, 2772000, 831600, 0, 12, 67452, 3993000, 20207880, 13305600, 665280
OFFSET
0,4
FORMULA
From Mélika Tebni, May 10 2023: (Start)
E.g.f. of column k: (x*(exp(x)-1))^k / k!.
Sum_{k=0..n-1} (-1)^(n+k-1)*T(n+k-1, k) = A000169(n), for n > 0. (End)
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 2;
[3] 0, 3;
[4] 0, 4, 12;
[5] 0, 5, 60;
[6] 0, 6, 210, 120;
[7] 0, 7, 630, 1260;
[8] 0, 8, 1736, 8400, 1680;
[9] 0, 9, 4536, 45360, 30240;
MAPLE
T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):
seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);
# second program:
egf := k-> (x*(exp(x)-1))^k / k!:
A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A362369(n, k), k=0..iquo(n, 2))), n=0..12); # Mélika Tebni, May 10 2023
PROG
(SageMath)
def A362369(n, k):
return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)
for n in range(10):
print([A362369(n, k) for k in range(n//2 + 1)])
CROSSREFS
Cf. A000169, A052506 (row sums), A362788, A362789.
Sequence in context: A357475 A066682 A239968 * A320311 A240140 A240141
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, May 04 2023
STATUS
approved