OFFSET
0,9
COMMENTS
REFERENCES
Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.
LINKS
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
Peter Luschny, The Bell transform
FORMULA
T(n, k) = n!*[t^k][x^n] (1 - x)^(t*(x - 1)).
T(n, k) = Sum_{j=k..n} (-1)^(n-k)*binomial(j, k)*k^(j-k)*Stirling1(n, j).
T(n, k) = Bell_{n, k}(a), where Bell_{n, k} is the partial Bell polynomial evaluated over the sequence a = {cfact(m) | m >= 0}, (see Mathematica).
T(n, k) = (-1)^(n-k)*t(n, k) where t(n, n) = 1 and t(n, k) = (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) for k > 0 and n > 0.
Let s(n) = (-1)^n*Sum_{k=1..n} (k-1)^(k-1)*T(n, k) for n >= 0, then s = A159075.
Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k this is A354796(n, k) for 0 <= k <= n and implies in particular for x = n >= 1 the identity Sum_{k=1..n} (k + n)^(k - 1)*T(n, k) = Gamma(2*n)/n! = A006963(n+1).
E.g.f. of column k >= 0: ((1 - t) * log(1 - t))^k / ((-1)^k * k!). - Werner Schulte, Jun 14 2022
EXAMPLE
Triangle T(n, k) begins:
[0] [1]
[1] [0, 1]
[2] [0, -1, 1]
[3] [0, -1, -3, 1]
[4] [0, -2, -1, -6, 1]
[5] [0, -6, 0, 5, -10, 1]
[6] [0, -24, 4, 15, 25, -15, 1]
[7] [0, -120, 28, 49, 35, 70, -21, 1]
[8] [0, -720, 188, 196, 49, 0, 154, -28, 1]
[9] [0, -5040, 1368, 944, 0, -231, -252, 294, -36, 1]
MAPLE
# The function BellMatrix is defined in A264428.
cfact := n -> ifelse(n = 0, 1, -(n - 1)!): BellMatrix(cfact, 10);
# Alternative:
t := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = n then 1 else (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) fi end:
T := (n, k) -> (-1)^(n-k)*t(n, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
# Using the e.g.f.:
egf := (1 - x)^(t*(x - 1)):
ser := series(egf, x, 11): coeffx := n -> coeff(ser, x, n):
row := n -> seq(n!*coeff(coeffx(n), t, k), k=0..n):
seq(print(row(n)), n = 0..9);
MATHEMATICA
cfact[n_] := If[n == 0, 1, -(n - 1)!];
R := Range[0, 10]; cf := Table[cfact[n], {n, R}];
Table[BellY[n, k, cf], {n, R}, {k, 0, n}] // Flatten
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jun 09 2022
STATUS
approved