Search: id:A135338
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%I A135338
%S A135338 1,1,1,1,3,1,2,7,6,1,6,20,25,10,1,24,76,105,65,15,1,120,364,511,385,140,
%T A135338 21,1,720,2108,2940,2401,1120,266,28,1
%V A135338 1,-1,1,1,-3,1,-2,7,-6,1,6,-20,25,-10,1,-24,76,-105,65,-15,1,120,-364,
               511,-385,140,-21,
%W A135338 1,-720,2108,-2940,2401,-1120,266,-28,1
%N A135338 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer 
               sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) 
               * (2+D) ] } where W is the Lambert W multi-valued function.
%C A135338 The lowering (or delta) operator for these polynomials is L = -1 + exp{ 
               2 + W[ -exp(-2) * (2+D) ] } = sum(j >= 1) A074059(j) * D^j / j! .
%C A135338 The raising operator is R = -x { 1 + W[ -exp(-2) * (2+D) ] } = x { 1 
               + sum(j >= 1) (-1)^j * PW(j-1,-2) * D^j / j! }, where PW(j-1,x) are 
               the polynomials of A042977.
%C A135338 W(x) here is W_-1 in the Monir reference and, about x = 0,
%C A135338 W[ -exp(-2) * (2+x) ] = -[ 2 + sum(j >= 1) (-1)^j * PW(j-1,-2) * x^j 
               / j! ] .
%C A135338 From the relation between delta and raising operators for
%C A135338 associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and 
               S
%C A135338 = (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74,
%C A135338 -593, -5298, ...) form a list partition transform pair (see A133314);
%C A135338 i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual
%C A135338 recursion relations. Applying Faa di Bruno's formula to L gives other
%C A135338 interesting integer relations between S and A074059.
%D A135338 F. Chapeau-Blondeau and A. Monir, Numerical Evaluation of the Lambert 
               W Function and Application to Generation of Generalized Gaussian 
               Noise With Exponent 1/2, IEEE Trans. on Signal Processing, Vol. 50, 
               No. 9, Sept. 2002, p. 2160-2164.
%F A135338 The row polynomials P(n,t) = sum(j=1,...,n) C(n,j) * t^j satisfy exp[P(.,
               t) * x] = exp{ -t * [(1+x) * ln(1+x) - 2x] }, with P(0,t) = 1 and 
               [ P(.,x) + P(.,y) ]^n = P(n,x+y) . See Mathworld and Wikipedia on 
               Sheffer sequences and umbral calculus for other general formulae, 
               including expansion theorems.
%Y A135338 Cf. A134685, A135494.
%Y A135338 Sequence in context: A134348 A074305 A151855 this_sequence A084602 A100888 
               A052914
%Y A135338 Adjacent sequences: A135335 A135336 A135337 this_sequence A135339 A135340 
               A135341
%K A135338 sign,tabl
%O A135338 1,5
%A A135338 Tom Copeland (tcjpn(AT)msn.com), Feb 15 2008

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