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
Examples
Triangle begins: 1; 1, 1; -1, 3, 1; 2, -1, 6, 1; -6, 0, 5, 10, 1; 24, 4, -15, 25, 15, 1; ...
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.
Crossrefs
Programs
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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
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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 *) -
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
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
Extensions
More terms from James Sellers, Jan 26 2001
Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007
Comments