A354794 - OEIS

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

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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)).

REFERENCES

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

LINKS

Table of n, a(n) for n=0..54.

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.

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)

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;

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

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)])

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).

Sequence in context: A327366 A327069 A327334 * A355401 A195596 A332054

Adjacent sequences: A354791 A354792 A354793 * A354795 A354796 A354797

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jun 09 2022

STATUS

approved