A008296 -id:A008296 - cp's OEIS frontend

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)!.

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

A185164 Coefficients of a set of polynomials associated with the derivatives of x^x.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 24, 40, 15, 120, 196, 105, 720, 1148, 700, 105, 5040, 7848, 5068, 1260, 40320, 61416, 40740, 12600, 945, 362880, 541728, 363660, 126280, 17325, 3628800, 5319072, 3584856, 1332100, 242550, 10395, 39916800, 57545280, 38764440, 15020720, 3213210, 270270

Offset: 2

Peter Bala, Mar 12 2012

Gould shows that the derivatives of x^x are given by (d/dx)^n(x^x) = (x^x)*Sum_{k = 0..n} (-1)^k*binomial(n,k)*(1 + log(x))^(n-k)*x^(-k)*R(k,x), where R(n,x) is a polynomial in x of degree floor(n/2). The first few values are R(0,x) = 1, R(1,x) = 0, R(2,x) = x, R(3,x) = x and R(4,x) = 2*x + 3*x^2. The coefficients of these polynomials are listed in the table for n >= 2. Gould gives an explicit formula for R(n,x) as a triple sum, and also an expression in terms of the Comtet numbers A008296.

This table read by diagonals gives A075856.

			Triangle begins
n\k.|.....1.....2.....3.....4
= = = = = = = = = = = = = = =
..2.|.....1
..3.|.....1
..4.|.....2.....3
..5.|.....6....10
..6.|....24....40....15
..7.|...120...196...105
..8.|...720..1148...700...105
..9.|..5040..7848..5068..1260
...
Fourth derivative of x^x:
x^(-x)*(d/dx)^4(x^x) = (1+log(x))^4 + C(4,2)/x^2*(1+log(x))^2*x - C(4,3)/x^3*(1+log(x)) + C(4,4)/x^4*(2*x + 3*x^2).
Example of recurrence relation for table entries:
T(7,2) = 4*T(6,2) + 6*T(5,1) = 4*40 + 6*6 = 196.

Robert Israel, Table of n, a(n) for n = 2..10001 (rows 2 to 200, flattened)
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
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.

Cf. A008296, A075856, A203852 (row sums).

T[2,1]:= 1:
for n from 3 to 15 do
  for k from 1 to floor(n/2) do
    T[n,k]:= (n-1-k)*`if`(k<= floor((n-1)/2),T[n-1,k],0) + `if`(k>=2 and k-1 <= floor((n-2)/2),(n-1)*T[n-2,k-1],0)
od od:
seq(seq(T[n,k],k=1..floor(n/2)),n=2..15); # Robert Israel, Jan 13 2016

m = 14; F = Exp[t (x + (1-x) Log[1-x])];
cc = CoefficientList[# + O[t]^m, t]& /@ CoefficientList[F + O[x]^m, x]* Range[0, m - 1]!;
Rest /@ Drop[cc, 2] (* Jean-François Alcover, Jun 26 2019 *)

# uses[bell_transform from A264428]
# Computes the full triangle for n>=0 and 0<=k<=n.
def A185164_row(n):
    g = lambda k: factorial(k-1) if k>0 else 0
    s = [g(k) for k in (0..n)]
    return bell_transform(n, s)
[A185164_row(n) for n in (0..10)] # Peter Luschny, Jan 13 2016

Recurrence relation: T(n+1,k) = (n - k)*T(n,k) + n*T(n-1,k-1).
The diagonal entries D(n,k) := T(n+k,k) satisfy the recurrence D(n+1,k) = n*D(n,k) + (n + k)*D(n,k-1) so this table read by diagonals is A075856.
E.g.f.: F(x,t) = exp(t*(x + (1 - x)*log(1 - x))) = Sum_{n = 0..oo} R(n,t)*x^n/n! = 1 + t*x^2/2! + t*x^3/3! + (2*t + 3*t^2)*x^4/4! + .... The e.g.f. F(x,t) satisfies the partial differential equation (1 - x)*dF/dx + t*dF/dt = x*t*F.
This gives the recurrence relation for the row generating polynomials: R(n+1,x) = n*R(n,x) - x*d/dx(R(n,x)) + n*x*R(n-1,x) for n >= 1, with initial conditions R(0,x) = 1, R(1,x) = 0.
The e.g.f. for the triangle read by diagonals is given by the series reversion (with respect to x) (x - t*(x + (1 - x)*log(1 - x)))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....
Diagonal sums: Sum_{k = 1..n} T(n+k,k) = n^n , n >= 1.
Row sums A203852.
Also the Bell transform of the sequence g(k) = (k-1)! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016

More terms from Jean-François Alcover, Jun 26 2019

A211193 E.g.f.: exp((1+x)^(1+x)-1).

Original entry on oeis.org

1, 1, 3, 10, 45, 221, 1315, 8324, 60809, 464113, 3993811, 35342814, 349085869, 3486862653, 38870528411, 429139127416, 5345350992113, 63994963427393, 887692696733827, 11284513262684914, 175285847038616301, 2298693217837384957, 40805829165456572691

Offset: 0

Robert G. Wilson v, Feb 03 2013

sign

Note that for odd n >= 31, a(n) is negative! - Vaclav Kotesovec, Feb 13 2013

Conjecture: For n > 1, a(n) == 1 (mod n). - Mélika Tebni, Aug 22 2021

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000110, A008296.

egf:= exp((1+x)^(1+x)-1);
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013
# second program: uses Lehmer-Comtet A008296.
A211193:= n-> add(combinat[bell](k)*A008296(n, k), k=1..n): A211193(0):=1:
seq(A211193(n), n=0..15); # Mélika Tebni, Aug 22 2021

Range[0, 22]! CoefficientList[ Series[ Exp[(1 + x)^(1 + x)], {x, 0, 22}], x]/E

x='x+O('x^66); Vec(serlaplace(exp((1+x)^(1+x)-1))) \\ Joerg Arndt, Nov 30 2014

E.g.f.: exp((1+x)^(1+x)-1).
a(n) ~ (n-2)! if n is even and a(n) ~ -(n-2)! if n is odd. - Vaclav Kotesovec, Feb 13 2013
a(n) = Sum_{k=1..n} Bell(k)*A008296(n, k) for n >= 1. - Mélika Tebni, Aug 22 2021

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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)!.

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A185164 Coefficients of a set of polynomials associated with the derivatives of x^x.

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A211193 E.g.f.: exp((1+x)^(1+x)-1).

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