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

1,5

Comments

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

References

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

Links

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.

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) {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 */
(Sage) # 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

Crossrefs

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.

Sequence in context: A131918 A010123 A039620 * A351397 A140185 A229341

Adjacent sequences: A008293 A008294 A008295 * A008297 A008298 A008299

Keyword

sign,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Jan 26 2001

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

Status

approved