OFFSET
0,2
COMMENTS
The first derivative of the Lambert W function is given by dW/dz = exp(-W)/(1+W). Further differentiation yields d^2/dz^2(W) = exp(-2*W)*(-2-W)/(1+W)^3, d^3/dz^3(W) = exp(-3*W)*(9+8*W+2*W^2)/(1+W)^5 and, in general, d^n/dz^n(W) = exp(-n*W)*R(n,W)/(1+W)^(2*n-1), where R(n,W) are the row polynomials of this triangle. - Peter Bala, Jul 22 2012
Conjecture: the polynomials have no real roots greater than or equal to -1. This is equivalent to the statement that the derivatives of the 0th branch of the Lambert W function have no real roots greater than -1/e. - Colin Linzer, Jan 29 2025
LINKS
G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
A. F. Beardon, Winding Numbers, Unwinding Numbers, and the Lambert W Function, Computational Methods and Function Theory, 2021.
George C. Greubel, On Szasz-Mirakyan-Jain Operators preserving exponential functions, arXiv:1805.06968 [math.CA], 2018.
Roy M. Howard, Schröder Based Series for the Lambert W Function, Curtin Univ. (Australia), ResearchGate (2025). See p. 8. See also On Schröder-Type Series Expansions for the Lambert W Function, AppliedMath (2025) Vol. 5, No. 2, Art. No. 66.
G. A. Kalugin and D. J. Jeffrey, Unimodal sequences show that Lambert is Bernstein, C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) pp. 50-56, 2011; arXiv:1011.5940 [math.CA], 2010.
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Lambert W-Function
FORMULA
E.g.f.: (LambertW(exp(x)*(x+y*(1+x)^2))-x)/(1+x). - Vladeta Jovovic, Nov 19 2003
a(n) = B(n)*(1+x)^(2*n-1), where B(1) = 1/(1+x), and for n>=2, B(n) = -(n!/(1+x)^n)*Sum_{m=1..n-1} (B(m)/m!)*Sum_{j=1..m} (-1)^(m-j)*binomial(m,j)*Sum_{i=0..n} j^(n-i)*binomial(j,i)*x^(m-i)/(n-i)!. - Vladimir Kruchinin, Apr 07 2011
Recurrence equation: T(n+1,k) = -n*T(n,k-1) - (3*n-k-1)*T(n,k) + (k+1)*T(n,k+1). - Peter Bala, Jul 22 2012
T(n,m) = Sum_{j=0..m} C(2*n+1,m-j)*(Sum_{k=0..j} (n+k+1)^(n+j)*(-1)^(n+k)/((j-k)!*k!)). - Vladimir Kruchinin, Feb 20 2018
EXAMPLE
Triangle begins:
n\k | 1 W W^2 W^3 W^4
==================================
1 | 1
2 | -2 -1
3 | 9 8 2
4 | -64 -79 -36 -6
5 | 625 974 622 192 24
...
T(5,2) = -4*(-79) - 9*(-36) + 3*(-6) = 622.
MAPLE
# After Vladimir Kruchinin, for 0 <= m <= n:
T := (n, m) -> add(add((-1)^(k+n)*binomial(j, k)*binomial(2*n+1, m-j)*(k+n+1)^(n+j), k=0..j)/j!, j=0..m): seq(seq(T(n, k), k=0..n), n=0..7); # Peter Luschny, Feb 23 2018
MATHEMATICA
Table[ Simplify[ (Evaluate[ D[ ProductLog[ z ], {z, n} ] ] /. ProductLog[ z ]->W)*z^n/W^n (1+W)^(2n-1) ], {n, 12} ] // TableForm
Flatten[ Table[ CoefficientList[ Simplify[ (Evaluate[D[ProductLog[z], {z, n}]] /. ProductLog[z] -> W) z^n / W^n (1 + W)^(2 n - 1)], W], {n, 8}]] (* Michael Somos, Jun 07 2012 *)
T[ n_, k_] := If[ n < 1 || k < 0, 0, Coefficient[ Simplify[(Evaluate[D[ProductLog[z], {z, n}]] /. ProductLog[z] -> W) z^n / W^n (1 + W)^(2 n - 1)], W, k]] (* Michael Somos, Jun 07 2012 *)
PROG
(Maxima)
B(n):=(if n=1 then 1/(1+x)*exp(-x) else -n!*sum((sum((-1)^(m-j)*binomial(m, j)*sum((j^(n-i)*binomial(j, i)*x^(m-i))/(n-i)!, i, 0, n), j, 1, m))*B(m)/m!, m, 1, n-1)/(1+x)^n);
a(n):=B(n)*(1+x)^(2*n-1);
/* Vladimir Kruchinin, Apr 07 2011 */
(Maxima)
a(n):=if n=1 then 1 else (n-1)!*(sum((binomial(n+k-1, n-1)*sum(binomial(k, j)*(x+1)^(n-j-1)*sum(binomial(j, l)*(-1)^(l)*sum((l^(n+j-i-1)*binomial(l, i)*x^(j-i))/(n+j-i-1)!, i, 0, l), l, 1, j), j, 1, k)), k, 1, n-1));
T(n, k):=coeff(ratsimp(a(n)), x, k);
for n: 1 thru 12 do print(makelist(T(n, k), k, 0, n-1));
/* Vladimir Kruchinin, Oct 09 2012 */
T(n, m):=sum(binomial(2*n+1, m-j)*sum(((n+k+1)^(n+j)*(-1)^(n+k))/((j-k)!*k!), k, 0, j), j, 0, m); /* Vladimir Kruchinin, Feb 20 2018 */
CROSSREFS
Cf. A054589.
AUTHOR
STATUS
approved