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A135338
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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.
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3
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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, 1, -720, 2108, -2940, 2401, -1120, 266, -28, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| 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! .
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.
W(x) here is W_-1 in the Monir reference and, about x = 0,
W[ -exp(-2) * (2+x) ] = -[ 2 + sum(j >= 1) (-1)^j * PW(j-1,-2) * x^j / j! ] .
From the relation between delta and raising operators for
associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and S
= (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74,
-593, -5298, ...) form a list partition transform pair (see A133314);
i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual
recursion relations. Applying Faa di Bruno's formula to L gives other
interesting integer relations between S and A074059.
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REFERENCES
| 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.
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LINKS
| P. Bala, Diagonals of triangles with generating function exp(t*F(x)).
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FORMULA
| 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.
From Peter Bala , Dec 09 2011: (Start)
E.g.f.: exp(t*(2*x-(1+x)*ln(1+x))) = 1 + t*x + (t^2-t)*x^2/2! + (t^3-3*t^2+t)*x^3/3! + ....
If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t (see the Bala link). The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1). In this case (x-t*(2*x-(1+x)*ln(1+x)))^(-1) = x/(1-t) - t/(1-t)^3*x^2/2! + (t+2*t^2)/(1-t)^5*x^3/3! - (2*t+6*t^2+7*t^3)/(1-t)^7*x^4/4! + .... So, for example, the (unsigned) third subdiagonal has o.g.f. (2*t+6*t^2+7*t^3)/(1-t)^7 = 2*t + 20*t^2 + 105*t^3 + 385*t^4 + ....
(End)
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CROSSREFS
| Cf. A134685, A135494.
Sequence in context: A074305 A151855 A186366 * A084602 A100888 A052914
Adjacent sequences: A135335 A135336 A135337 * A135339 A135340 A135341
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KEYWORD
| sign,tabl
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AUTHOR
| Tom Copeland (tcjpn(AT)msn.com), Feb 15 2008
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