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A354794
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Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.
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9
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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
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OFFSET
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0,8
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COMMENTS
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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)).
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REFERENCES
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Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.
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LINKS
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FORMULA
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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).
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)
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EXAMPLE
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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;
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MAPLE
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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)):
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MATHEMATICA
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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
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PROG
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(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)])
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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