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A042977
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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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14
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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# 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
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MATHEMATICA
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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 *)
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PROG
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(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);
(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));
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 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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