A345651 -id:A345651 - cp's OEIS frontend

A008296 Triangle of Lehmer-Comtet numbers of the first kind.

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800

Offset: 1

N. J. A. Sloane

Triangle arising in the expansion of ((1+x)*log(1+x))^n.

Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			Triangle begins:
   1;
   1,  1;
  -1,  3,   1;
   2, -1,   6,  1;
  -6,  0,   5, 10,  1;
  24,  4, -15, 25, 15, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
Tian-Xiao He and Yuanziyi Zhang, Centralizers of the Riordan Group, arXiv:2105.07262 [math.CO], 2021.
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
Diagonals: A000142, A000217, A059302.
Cf. A176118.

for n from 1 to 20 do for k from 1 to n do
printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
# second program:
A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # Mélika Tebni, Aug 22 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1,k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* Jean-François Alcover, Apr 29 2011 *)

{T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* Michael Somos, Nov 15 2002 */

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # Peter Luschny, Jan 16 2016

E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - Len Smiley
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
From Peter Bala, Mar 14 2012: (Start)
E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
(End)
Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - Alois P. Heinz, Aug 25 2021

More terms from James Sellers, Jan 26 2001

Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007

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)!.

Original entry on oeis.org

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, 1, 0, -120, 28, 49, 35, 70, -21, 1, 0, -720, 188, 196, 49, 0, 154, -28, 1, 0, -5040, 1368, 944, 0, -231, -252, 294, -36, 1, 0, -40320, 11016, 5340, -820, -1365, -987, -1050, 510, -45, 1

Offset: 0

Peter Luschny, Jun 09 2022

The triangle is the matrix inverse of the Bell transform of n^n (A354794).
The numbers (-1)^(n-k)*T(n, k) are known as the Lehmer-Comtet numbers of 1st kind (A008296).
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).

			Triangle T(n, k) begins:
[0] [1]
[1] [0,     1]
[2] [0,    -1,    1]
[3] [0,    -1,   -3,   1]
[4] [0,    -2,   -1,  -6,   1]
[5] [0,    -6,    0,   5, -10,    1]
[6] [0,   -24,    4,  15,  25,  -15,    1]
[7] [0,  -120,   28,  49,  35,   70,  -21,   1]
[8] [0,  -720,  188, 196,  49,    0,  154, -28,   1]
[9] [0, -5040, 1368, 944,   0, -231, -252, 294, -36, 1]

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

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

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.

# The function BellMatrix is defined in A264428.
cfact := n -> ifelse(n = 0, 1, -(n - 1)!): BellMatrix(cfact, 10);
# Alternative:
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:
T := (n, k) -> (-1)^(n-k)*t(n, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
# Using the e.g.f.:
egf := (1 - x)^(t*(x - 1)):
ser := series(egf, x, 11): coeffx := n -> coeff(ser, x, n):
row := n -> seq(n!*coeff(coeffx(n), t, k), k=0..n):
seq(print(row(n)), n = 0..9);

cfact[n_] := If[n == 0, 1, -(n - 1)!];
R := Range[0, 10]; cf := Table[cfact[n], {n, R}];
Table[BellY[n, k, cf], {n, R}, {k, 0, n}] // Flatten

T(n, k) = n!*[t^k][x^n] (1 - x)^(t*(x - 1)).
T(n, k) = Sum_{j=k..n} (-1)^(n-k)*binomial(j, k)*k^(j-k)*Stirling1(n, j).
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).
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.
Let s(n) = (-1)^n*Sum_{k=1..n} (k-1)^(k-1)*T(n, k) for n >= 0, then s = A159075.
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).
E.g.f. of column k >= 0: ((1 - t) * log(1 - t))^k / ((-1)^k * k!). - Werner Schulte, Jun 14 2022

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A008296 Triangle of Lehmer-Comtet numbers of the first kind.

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