A354794 Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 27, 19, 6, 1, 0, 256, 175, 55, 10, 1, 0, 3125, 2101, 660, 125, 15, 1, 0, 46656, 31031, 9751, 1890, 245, 21, 1, 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1, 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1

Offset: 0

Author

Peter Luschny, Jun 09 2022

Keywords

• nonn
• tabl

Comments

For the definition of the Bell transform see A264428. The Bell transform of {(-m)^m | m >= 0} is A039621. The numbers A039621(n, k) are known as the Lehmer-Comtet numbers of 2nd kind. We think it is more natural to use Bell_{n, k}({m^m}) as the basis for the definition (and let the triangle start at (0, 0)).

Examples

			Triangle T(n, k) begins:
[0] 1;
[1] 0,        1;
[2] 0,        1,        1;
[3] 0,        4,        3,       1;
[4] 0,       27,       19,       6,      1;
[5] 0,      256,      175,      55,     10,     1;
[6] 0,     3125,     2101,     660,    125,    15,    1;
[7] 0,    46656,    31031,    9751,   1890,   245,   21,   1;
[8] 0,   823543,   543607,  170898,  33621,  4550,  434,  28,  1;
[9] 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1;
		

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
• Wikipedia, Bell polynomials.

Crossrefs

Cf. A264428, A039621 (signed variant), A195979 (row sums), A000312 (column 1), A045531 (column 2), A281596 (column 3), A281595 (column 4), A000217 (diagonal 1), A215862 (diagonal 2), A354795 (matrix inverse), A137452 (Abel).

Programs

Maple

T := (n, k) -> if n = k then 1 else
add((-1)^j*(n-j-1)^(n-1)/(j!*(k-1-j)!), j = 0.. k-1) fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);
# Alternatively, using the function BellMatrix from A264428:
BellMatrix(n -> n^n, 9);
# Or by recursion:
R := proc(n, k, m) option remember;
   if k < 0 or n < 0 then 0 elif k = 0 then 1 else
   m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
A039621 := (n, k) -> ifelse(n = 0, 1, R(k-1, n-k, n-k)):

Mathematica

Unprotect[Power]; Power[0, 0] = 1; pow[n_] := n^n;
R = Range[0, 9]; T[n_, k_] := BellY[n, k, pow[R]];
Table[T[n, k], {n, R}, {k, 0, n}] // Flatten

Python

from functools import cache
@cache
def t(n, k, m):
    if k < 0 or n < 0: return 0
    if k == 0: return n ** k
    return m * t(n, k - 1, m) + t(n - 1, k, m + 1)
def A354794(n, k): return t(k - 1, n - k, n - k) if n != k else 1
for n in range(9): print([A354794(n, k) for k in range(n + 1)])

Formula

T(n, k) = Bell_{n, k}(A000312), where Bell_{n, k} is the partial Bell polynomial evaluated over the powers m^m (with 0^0 = 1). See the Mathematica program.

T(n, k) = Sum_{j=0..k-1} (-1)^j*(n-j-1)^(n - 1)/(j! * (k-1-j)!) for 0 <= k < n and T(n, n) = 1.

T(n, k) = r(k-1, n-k, n-k) for n,k >= 1 and T(0, 0) = 1, where r(n, k, m) = m*r(n, k-1, m) + r(n-1, k, m+1) and r(n, 0, m) = 1. (see Vladimir Kruchinin's formula in A039621).

Sum_{k=1..n} binomial(k + x - 1, k-1)*(k-1)!*T(n, k) = (n + x)^(n - 1) for n >= 1.

Sum_{k=1..n} (-1)^(k+j)*Stirling1(k, j)*T(n, k) = n^(n-j)*binomial(n-1, j-1) for n >= 1, which are, up to sign, the coefficients of the Abel polynomials (A137452).

From Werner Schulte, Jun 14 2022 and Jun 19 2022: (Start)

E.g.f. of column k >= 0: (Sum_{i>0} (i-1)^(i-1) * t^i / i!)^k / k!.

Conjecture: T(n, k) = Sum_{i=0..n-k} A048994(n-k, i) * A048993(n+i-1, n-1) for 0 < k <= n and T(n, 0) = 0^n for n >= 0; proved by Mike Earnest, see link at A354797. (End)