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A042977 Triangle T(n,k) read by rows: coefficients of a polynomial sequence occurring when calculating the n-th derivative of Lambert function W.

%I A042977 #109 Jul 20 2025 15:03:38
%S A042977 1,-2,-1,9,8,2,-64,-79,-36,-6,625,974,622,192,24,-7776,-14543,-11758,
%T A042977 -5126,-1200,-120,117649,255828,248250,137512,45756,8640,720,-2097152,
%U A042977 -5187775,-5846760,-3892430,-1651480,-445572,-70560,-5040
%N A042977 Triangle T(n,k) read by rows: coefficients of a polynomial sequence occurring when calculating the n-th derivative of Lambert function W.
%C A042977 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
%C A042977 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
%H A042977 G. C. Greubel, <a href="/A042977/b042977.txt">Table of n, a(n) for the first 75 rows, flattened</a>
%H A042977 A. F. Beardon, <a href="https://doi.org/10.1007/s40315-021-00398-1">Winding Numbers, Unwinding Numbers, and the Lambert W Function</a>, Computational Methods and Function Theory, 2021.
%H A042977 George C. Greubel, <a href="https://arxiv.org/abs/1805.06968">On Szasz-Mirakyan-Jain Operators preserving exponential functions</a>, arXiv:1805.06968 [math.CA], 2018.
%H A042977 Roy M. Howard, <a href="https://doi.org/10.13140/RG.2.2.12452.39046">Schröder Based Series for the Lambert W Function</a>, Curtin Univ. (Australia), ResearchGate (2025). See p. 8. See also <a href="https://doi.org/10.3390/appliedmath5020066">On Schröder-Type Series Expansions for the Lambert W Function</a>, AppliedMath (2025) Vol. 5, No. 2, Art. No. 66.
%H A042977 G. A. Kalugin and D. J. Jeffrey, <a href="https://arxiv.org/abs/1011.5940">Unimodal sequences show that Lambert is Bernstein</a>, C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) pp. 50-56, 2011; arXiv:1011.5940 [math.CA], 2010.
%H A042977 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1104.5065">Derivation of Bell Polynomials of the Second Kind</a>, arXiv:1104.5065 [math.CO], 2011.
%H A042977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%F A042977 E.g.f.: (LambertW(exp(x)*(x+y*(1+x)^2))-x)/(1+x). - _Vladeta Jovovic_, Nov 19 2003
%F A042977 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
%F A042977 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
%F A042977 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
%e A042977 Triangle begins:
%e A042977  n\k |    1    W   W^2   W^3   W^4
%e A042977 ==================================
%e A042977   1  |    1
%e A042977   2  |   -2   -1
%e A042977   3  |    9    8     2
%e A042977   4  |  -64  -79   -36    -6
%e A042977   5  |  625  974   622   192    24
%e A042977 ...
%e A042977 T(5,2) = -4*(-79) - 9*(-36) + 3*(-6) = 622.
%p A042977 # After Vladimir Kruchinin, for 0 <= m <= n:
%p A042977 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
%t A042977 Table[ Simplify[ (Evaluate[ D[ ProductLog[ z ], {z, n} ] ] /. ProductLog[ z ]->W)*z^n/W^n (1+W)^(2n-1) ], {n, 12} ] // TableForm
%t A042977 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 A042977 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 *)
%o A042977 (Maxima)
%o A042977 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);
%o A042977 a(n):=B(n)*(1+x)^(2*n-1);
%o A042977 /* _Vladimir Kruchinin_, Apr 07 2011 */
%o A042977 (Maxima)
%o A042977 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));
%o A042977 T(n, k):=coeff(ratsimp(a(n)), x, k);
%o A042977 for n: 1 thru 12 do print(makelist(T(n, k), k, 0, n-1));
%o A042977 /* _Vladimir Kruchinin_, Oct 09 2012 */
%o A042977 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 */
%Y A042977 Cf. A013703 (twice row sums), A000444, A000525, A064781, A064785, A064782.
%Y A042977 First column A000169, main diagonal A000142, first subdiagonal A052582.
%Y A042977 Cf. A054589.
%K A042977 sign,tabl,look
%O A042977 0,2
%A A042977 _Wouter Meeussen_