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A042977
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Triangle T(n,k) read by rows: coefficients of n-th derivative of Lambert function W expanded in powers of W.
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11
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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LINKS
| Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065
Eric Weisstein's World of Mathematics, Lambert W-Function
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FORMULA
| E.g.f.: (LambertW(exp(x)*(x+y*(1+x)^2))-x)/(1+x). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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). [From Vladimir Kruchinin kru(AT)ie.tusur.ru, Apr 07 2011]
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EXAMPLE
| Triangle begins:
1;
-2 - 1 W;
9 + 8 W + 2 W^2;
-64 - 79 W - 36 W^2 - 6 W^3;
625 + 974 W + 622 W^2 + 192 W^3 + 24 W^4; ...
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MATHEMATICA
| Table[ Simplify[ (Evaluate[ D[ ProductLog[ z ], {z, n} ] ]/ .ProductLog[ z ]->W)*z^n/W^n (1+W)^(2n-1) ], {n, 12} ];
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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); [From Vladimir Kruchinin kru(AT)ie.tusur.ru, Apr 07 2011]
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CROSSREFS
| Cf. A013703 (twice row sums), A000444, A000525, A064781, A064785, A064782.
First column A000169, main diagonal A000142, first subdiagonal A052582.
Sequence in context: A133169 A133175 A188659 * A198106 A108290 A108291
Adjacent sequences: A042974 A042975 A042976 * A042978 A042979 A042980
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KEYWORD
| sign,tabl
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be)
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