A354795 Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.

1, 0, 1, 0, -1, 1, 0, -1, -3, 1, 0, -2, -1, -6, 1, 0, -6, 0, 5, -10, 1, 0, -24, 4, 15, 25, -15, 1, 0, -120, 28, 49, 35, 70, -21, 1, 0, -720, 188, 196, 49, 0, 154, -28, 1, 0, -5040, 1368, 944, 0, -231, -252, 294, -36, 1, 0, -40320, 11016, 5340, -820, -1365, -987, -1050, 510, -45, 1

Offset

0,9

Comments

The triangle is the matrix inverse of the Bell transform of n^n (A354794).

The numbers (-1)^(n-k)*T(n, k) are known as the Lehmer-Comtet numbers of 1st kind (A008296).

The function cfact is the 'complementary factorial' (name is ad hoc) and written \hat{!} in TeX mathmode. 1/(cfact(-n) * cfact(n)) = signum(-n) * n for n != 0. It is related to the Roman factorial (A159333). The Bell transform of the factorial are the Stirling cycle numbers (A132393).

References

Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.

Links

Table of n, a(n) for n=0..65.

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

Cf. A354794 (matrix inverse), A176118 (row sums), A005727 (alternating row sums), A045406 (column 2), A347276 (column 3), A345651 (column 4), A298511 (central), A008296 (variant), A159333, A264428, A159075, A006963, A354796.

Sequence in context: A307752 A101548 A117430 * A143676 A002726 A119734

Adjacent sequences: A354792 A354793 A354794 * A354796 A354797 A354798

Keyword

sign,tabl

Author

Peter Luschny, Jun 09 2022

Status

approved