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A354795 Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.

%I A354795 #34 Jun 15 2022 01:14:14
%S A354795 1,0,1,0,-1,1,0,-1,-3,1,0,-2,-1,-6,1,0,-6,0,5,-10,1,0,-24,4,15,25,-15,
%T A354795 1,0,-120,28,49,35,70,-21,1,0,-720,188,196,49,0,154,-28,1,0,-5040,
%U A354795 1368,944,0,-231,-252,294,-36,1,0,-40320,11016,5340,-820,-1365,-987,-1050,510,-45,1
%N A354795 Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.
%C A354795 The triangle is the matrix inverse of the Bell transform of n^n (A354794).
%C A354795 The numbers (-1)^(n-k)*T(n, k) are known as the Lehmer-Comtet numbers of 1st kind (A008296).
%C A354795 The function cfact is the 'complementary factorial' (name is ad hoc) and written \hat{!} in TeX mathmode. 1/(cfact(-n) * cfact(n)) = signum(-n) * n for n != 0. It is related to the Roman factorial (A159333). The Bell transform of the factorial are the Stirling cycle numbers (A132393).
%D A354795 Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.
%H A354795 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 A354795 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%F A354795 T(n, k) = n!*[t^k][x^n] (1 - x)^(t*(x - 1)).
%F A354795 T(n, k) = Sum_{j=k..n} (-1)^(n-k)*binomial(j, k)*k^(j-k)*Stirling1(n, j).
%F A354795 T(n, k) = Bell_{n, k}(a), where Bell_{n, k} is the partial Bell polynomial evaluated over the sequence a = {cfact(m) | m >= 0}, (see Mathematica).
%F A354795 T(n, k) = (-1)^(n-k)*t(n, k) where t(n, n) = 1 and t(n, k) = (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) for k > 0 and n > 0.
%F A354795 Let s(n) = (-1)^n*Sum_{k=1..n} (k-1)^(k-1)*T(n, k) for n >= 0, then s = A159075.
%F A354795 Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k this is A354796(n, k) for 0 <= k <= n and implies in particular for x = n >= 1 the identity Sum_{k=1..n} (k + n)^(k - 1)*T(n, k) = Gamma(2*n)/n! = A006963(n+1).
%F A354795 E.g.f. of column k >= 0: ((1 - t) * log(1 - t))^k / ((-1)^k * k!). - _Werner Schulte_, Jun 14 2022
%e A354795 Triangle T(n, k) begins:
%e A354795 [0] [1]
%e A354795 [1] [0,     1]
%e A354795 [2] [0,    -1,    1]
%e A354795 [3] [0,    -1,   -3,   1]
%e A354795 [4] [0,    -2,   -1,  -6,   1]
%e A354795 [5] [0,    -6,    0,   5, -10,    1]
%e A354795 [6] [0,   -24,    4,  15,  25,  -15,    1]
%e A354795 [7] [0,  -120,   28,  49,  35,   70,  -21,   1]
%e A354795 [8] [0,  -720,  188, 196,  49,    0,  154, -28,   1]
%e A354795 [9] [0, -5040, 1368, 944,   0, -231, -252, 294, -36, 1]
%p A354795 # The function BellMatrix is defined in A264428.
%p A354795 cfact := n -> ifelse(n = 0, 1, -(n - 1)!): BellMatrix(cfact, 10);
%p A354795 # Alternative:
%p A354795 t := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = n then 1 else (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) fi end:
%p A354795 T := (n, k) -> (-1)^(n-k)*t(n, k):
%p A354795 seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
%p A354795 # Using the e.g.f.:
%p A354795 egf := (1 - x)^(t*(x - 1)):
%p A354795 ser := series(egf, x, 11): coeffx := n -> coeff(ser, x, n):
%p A354795 row := n -> seq(n!*coeff(coeffx(n), t, k), k=0..n):
%p A354795 seq(print(row(n)), n = 0..9);
%t A354795 cfact[n_] := If[n == 0, 1, -(n - 1)!];
%t A354795 R := Range[0, 10]; cf := Table[cfact[n], {n, R}];
%t A354795 Table[BellY[n, k, cf], {n, R}, {k, 0, n}] // Flatten
%Y A354795 Cf. A354794 (matrix inverse), A176118 (row sums), A005727 (alternating row sums), A045406 (column 2), A347276 (column 3), A345651 (column 4), A298511 (central), A008296 (variant), A159333, A264428, A159075, A006963, A354796.
%K A354795 sign,tabl
%O A354795 0,9
%A A354795 _Peter Luschny_, Jun 09 2022