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%I #32 Jun 19 2022 16:21:12
%S 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,
%T 125,15,1,0,46656,31031,9751,1890,245,21,1,0,823543,543607,170898,
%U 33621,4550,434,28,1,0,16777216,11012415,3463615,688506,95781,9702,714,36,1
%N Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.
%C 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)).
%D Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1216/RMJ-1985-15-2-461">Numbers Associated with Stirling Numbers and x^x</a>, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bell_polynomials">Bell polynomials</a>.
%F 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.
%F 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.
%F 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).
%F Sum_{k=1..n} binomial(k + x - 1, k-1)*(k-1)!*T(n, k) = (n + x)^(n - 1) for n >= 1.
%F 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).
%F From _Werner Schulte_, Jun 14 2022 and Jun 19 2022: (Start)
%F E.g.f. of column k >= 0: (Sum_{i>0} (i-1)^(i-1) * t^i / i!)^k / k!.
%F 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)
%e Triangle T(n, k) begins:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 1, 1;
%e [3] 0, 4, 3, 1;
%e [4] 0, 27, 19, 6, 1;
%e [5] 0, 256, 175, 55, 10, 1;
%e [6] 0, 3125, 2101, 660, 125, 15, 1;
%e [7] 0, 46656, 31031, 9751, 1890, 245, 21, 1;
%e [8] 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1;
%e [9] 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1;
%p T := (n, k) -> if n = k then 1 else
%p add((-1)^j*(n-j-1)^(n-1)/(j!*(k-1-j)!), j = 0.. k-1) fi:
%p seq(seq(T(n, k), k = 0..n), n = 0..9);
%p # Alternatively, using the function BellMatrix from A264428:
%p BellMatrix(n -> n^n, 9);
%p # Or by recursion:
%p R := proc(n, k, m) option remember;
%p if k < 0 or n < 0 then 0 elif k = 0 then 1 else
%p m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
%p A039621 := (n, k) -> ifelse(n = 0, 1, R(k-1, n-k, n-k)):
%t Unprotect[Power]; Power[0, 0] = 1; pow[n_] := n^n;
%t R = Range[0, 9]; T[n_, k_] := BellY[n, k, pow[R]];
%t Table[T[n, k], {n, R}, {k, 0, n}] // Flatten
%o (Python)
%o from functools import cache
%o @cache
%o def t(n, k, m):
%o if k < 0 or n < 0: return 0
%o if k == 0: return n ** k
%o return m * t(n, k - 1, m) + t(n - 1, k, m + 1)
%o def A354794(n, k): return t(k - 1, n - k, n - k) if n != k else 1
%o for n in range(9): print([A354794(n, k) for k in range(n + 1)])
%Y 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).
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Jun 09 2022
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