OFFSET
0,8
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)).
T(n+1, k) is the number of partitions of 2*n-k labeled balls into a set of n subsets, by allocating the first n-k into their own subsets, then distributing the n remaining balls such that the n-k remaining subsets are nonempty. - Natalia L. Skirrow, Jan 03 2026
REFERENCES
Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.
LINKS
Andrei Z. Broder, The r-Stirling numbers, Discrete Mathematics vol. 49 iss. 3 pp. 241-259, 1984.
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.
Peter Luschny, Examples of the Bell transform, a SageMath notebook.
Math StackExchange, Is there a way to simplify this closed form for Stirling numbers of the second kind?, 2025.
Wikipedia, Bell polynomials.
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)
From Natalia L. Skirrow, Jan 03 2026: (Start)
As an r-subset number, T(n+1, k) = Stirling2(n-k; 2*n-k, n).
Sum_{i=0..m} Stirling2(n+i, k+m)*Stirling1(m, i) = Sum_{j=0..n-k} C(n-m-k, j) * C(k+j, j) * j! * T(n+1, k+j+1) * (-1)^j.
Special case: Stirling2(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k+j, j)*j!*T(n+1, k+j+1)*(-1)^j. (End)
Conjectures from Mikhail Kurkov, Apr 20 2026: (Start)
T(n, k) = k * T(n, k+1) + T(n-1, k-1) + (n-k-1) * T(n-1, k) with T(n, 0) = 0^n, T(n, k) = 0 for k < 0 or k > n.
T(n, k) = Sum_{j=0..n-k} binomial(n+j-1, k-1) * A075856(n-k, j).
T(n+1, k+1) = Sum_{j=k..n} T(n, j) * A291978(j, k). (End)
From Peter Luschny, May 08 2026: (Start)
T(n, k) = (n!/k!) * [z^n] (1 - exp(-T(z)))^k where T(z) is the Cayley tree EGF (A000169).
The row polynomials P(n, x) are given by Sum_{n>=0} P(n, x) * z^n/n! = exp(x*(1 - exp(W(-z)))) where W denotes Lambert's W function.
P(n + 1, x) = x * Sum_{k=0..n} binomial(n, k) * k^k * P(n - k, x) = x * Sum_{k=0..n} A395701(n, k) * P(n - k, x) is a recurrence for the row polynomials starting with P(0, x) = 1. From this recurrence the recurrence of the coefficients given above by Mikhail Kurkov can be derived. (End)
From Peter Luschny, May 10 2026: (Start)
The rows of the array T(n + k, k) are given by linear recurrences with recurrence coefficients RCrow(n) = [(-1)^(j + 1)*binomial(2*n + 1, j), j = 1..2*n+1].
EXAMPLE
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;
.
Seen as an array A(n, k) = T(n + k, k):
[0] 1, 1, 1, 1, 1, 1, ... A000012
[1] 0, 1, 3, 6, 10, 15, ... A000217
[2] 0, 4, 19, 55, 125, 245, ... A215862
[3] 0, 27, 175, 660, 1890, 4550, ... A395908
[4] 0, 256, 2101, 9751, 33621, 95781, ...
[5] 0, 3125, 31031, 170898, 688506, 2263065, ...
[6] 0, 46656, 543607, 3463615, 15958405, 59411605, ...
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);
# Alternative: using the function BellMatrix from A264428:
BellMatrix(n -> n^n, 9);
# Alternative:
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)):
# Alternative: using the EGF.
N := 9: x := 'x': z := 'z': T := series(-LambertW(-z), z, N + 1):
F := exp(x*(1 - exp(-T))): EGF := series(F, z, N+1):
A354794row := n -> seq(coeff(simplify(coeff(EGF, z, n) * n!), x, k), k = 0..n):
seq(A354794row(n), n = 0..N); # Peter Luschny, May 08 2026
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
PROG
(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)])
(Python)
from math import factorial
def ArrayRow(n: int, size: int) -> list[int]:
if size <= 0: return []
row = [1 if n == 0 else 0]
for k in range(1, size):
total = 0; comb = 1
for j in range(k):
term = comb * pow(n + k - j - 1, n + k - 1)
total += term if j % 2 == 0 else -term
comb = comb * (k - 1 - j) // (j + 1)
row.append(total // factorial(k - 1))
return row
for n in range(8): print([n], ArrayRow(n, 9)) # Peter Luschny, May 10 2026
(PARI) \\ using function bell_matrix from A264428
bell_matrix(n -> n^n, 9) \\ Peter Luschny, May 07 2026
(PARI) inverse_bell_matrix_row(n, f) = { my(A, v1 = vector(n-1, i, f(i)), v2 = vector(n, i, 0)); v2[1] = 1; for(i=2, n, A = 1; v2[i] = sum(j=1, i-1, A *= -(i-n-j)/j; ((1-n)*A - (i-n)*A/(j+1)) * v2[i-j] * v1[j]) / (i-1)); Vecrev(v2) }
row(n) = if(n==0, [1], concat(0, inverse_bell_matrix_row(n, x->-(x-1)!))) \\ Mikhail Kurkov, May 08 2026
(PARI) bell_matrix_column(n, k, f) = { my(A, v1 = vector(n-1, i, f(i)), v2 = vector(n, i, 0)); v2[1] = 1; for(i=2, n, A = 1; v2[i] = sum(j=1, i-1, A *= (i+k-j)/j; ((k+1)*A - (i+k)*A/(j+1)) * v2[i-j] * v1[j]) / (i-1)); v2 } \\ computes first n elements of the k-th column starting from the first nonzero element (equals 1)
column(n, k) = bell_matrix_column(n, k, x->x^x) \\ Mikhail Kurkov, May 09 2026
(SageMath)
def A354794triangle(size: int) -> list[list[int]]:
R.<x> = PolynomialRing(ZZ)
mm = [1] + [m^m for m in range(1, size+1)]
P = [R(1)]
for n in range(size):
Qrow = [0]*(n + 1) # A395701
binom = 1
for m in range(n+1):
if m > 0: binom = binom * (n - m + 1) // m
Qrow[m] = binom * mm[m]
P.append(x * sum(Qrow[m] * P[n-m] for m in range(n+1)))
return [p.list() for p in P]
for row in A354794triangle(9): print(row) # Peter Luschny, May 08 2026
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Jun 09 2022
STATUS
approved