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A008296 Triangle of Lehmer-Comtet numbers of first kind. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Triangle arising in 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.

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_{l} binomial(l, k)*k^(l-k)*stirling1(n, l).

From Peter Bala, Mar 14 2012: (Start)

E.g.f.: exp(t*(1 + x)*log(1 + x)) = sum {n = 0..inf} 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)

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

with(combinat): for n from 1 to 20 do for k from 1 to n do printf(`%d, `, sum(binomial(l, k)*k^(l-k)*stirling1(n, l), l=k..n)) od: od:

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-Fran├žois 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)

# The function bell_matrix is defined in 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, A185164.

Diagonals give A000142, A045406, A000217, A059302.

Row sums give A005727.

T(2n,n) give A298511.

Sequence in context: A131918 A010123 A039620 * A140185 A229341 A106790

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

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)