A000525 -id:A000525 - cp's OEIS frontend

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

1, -2, -1, 9, 8, 2, -64, -79, -36, -6, 625, 974, 622, 192, 24, -7776, -14543, -11758, -5126, -1200, -120, 117649, 255828, 248250, 137512, 45756, 8640, 720, -2097152, -5187775, -5846760, -3892430, -1651480, -445572, -70560, -5040

Offset: 0

Wouter Meeussen

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

			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.

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

Cf. A013703 (twice row sums), A000444, A000525, A064781, A064785, A064782.
First column A000169, main diagonal A000142, first subdiagonal A052582.
Cf. A054589.

# 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

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 *)

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 */

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 */

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

A008295 Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 5, 9, 9, 4, 13, 34, 64, 64, 9, 35, 119, 326, 625, 625, 20, 95, 401, 1433, 4016, 7776, 7776, 48, 262, 1316, 5799, 21256, 60387, 117649, 117649, 115, 727, 4247, 22224, 100407, 373895, 1071904, 2097152, 2097152

Offset: 0

N. J. A. Sloane

T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - Sean A. Irvine, Mar 22 2018

			Triangle begins with T(0,0):
n\k 0  1   2    3    4    5    6
0   1
1   1  1
2   1  2   2
3   2  5   9    9
4   4 13  34   64   64
5   9 35 119  326  625  625
6  20 95 401 1433 4016 7776 7776

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

Sean A. Irvine, Table of n, a(n) for n = 0..527
J. Riordan, The numbers of labeled colored and chromatic trees, Acta Mathematica 97 (1957) 211.
Index entries for sequences related to rooted trees

Main diagonal is A000169.
Columns k=0..6 are A000081, A000107, A000524, A000444, A000525, A064781, A064782.
Cf. A034799.

m = 9; r[_] = 0;
Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
(CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* Jean-François Alcover, Oct 23 2019 *)

E.g.f.: r(x,y) = T(n,k) * y^k * x^n / k! satisfies r(x,y) * exp(r(x)) = (1+y) * r(x) * exp(r(x,y)) where r(x) is the o.g.f. for A000081. - Sean A. Irvine, Mar 22 2018

More terms from Sean A. Irvine, Mar 22 2018

Name edited by Andrew Howroyd, Mar 23 2023

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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.

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