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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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
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

LINKS

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened

D. J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, Preprint 2015.

George C. Greubel, On Szasz-Mirakyan-Jain Operators preserving exponential functions, arXiv:1805.06968 [math.CA], 2018.

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!*sum(m=1..n-1, (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)!)))*B(m)/m!)/(1+x)^n). - 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. A013703 (twice row sums), A000444, A000525, A064781, A064785, A064782.

First column A000169, main diagonal A000142, first subdiagonal A052582.

Cf. A054589.

Sequence in context: A133169 A133175 A188659 * A198106 A243261 A230358

Adjacent sequences:  A042974 A042975 A042976 * A042978 A042979 A042980

KEYWORD

sign,tabl,look

AUTHOR

Wouter Meeussen

STATUS

approved

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Last modified January 22 18:06 EST 2019. Contains 319365 sequences. (Running on oeis4.)