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A319201
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Expansion of f(t) = F^{[-1]}(t)/t, where F(x) = x/(1 - x^2 - x^3).
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5
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1, 0, -1, -1, 2, 5, -2, -21, -14, 72, 138, -165, -803, -143, 3575, 4732, -11674, -36244, 15130, 195738, 152456, -799102, -1700272, 2042975, 11038183, 2582670, -53547795, -76684530, 185864265, 618689190, -231325605, -3506922585, -2974386450, 14866619160, 33459332610, -38401746930, -223156727472
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OFFSET
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0,5
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COMMENTS
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The compositional inverse F^{[-1]} of F(x) = x/(1 - x^2 - x^3) (appearing in the Riordan matrix of the Bell type R = (F(x)/x, F(x)) from A104578) is needed for the inverse matrix of R, the Riordan matrix R^(-1) = (f(t), t*f(t)).
This function f(t) is also needed for the A- and Z-sequences of this Riordan matrix R, namely A(n) = [t^n](1/f(t)) = A319202(n) and Z(n) = [t^n]((1/t)*(1/f(t) - 1)) = A319202(n+1).
For the expansion of the compositional inverse of x/(1 + x^2 + x^3) see A001005.
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LINKS
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FORMULA
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a(n) = [t^n] G(t), where G(t) = F^{[-1]}(t)/t, and F(x) = x/(1 - x^2 - x^3).
O.g.f.: G(t) can be computed by Lagrange inversion of F.
a(n) = b(n+1)/(n+1), for n >= 0, where b(n+1) is the coefficient of x^n of (1 - x^2 - x^3)^(n+1). This follows from the Lagrange inversion series for G(x) = F^{[-1]}(x)/x.
a(n) = (1/(n+1))*(Sum_{2*e2 + 3*e3 = n} (-1)^{e2 + e3}*(n+1)!/(n+1 - (e2 + e3))!*e2!*e3!) (from the multinomial formula for (x1 + x2 + x3)^(n+1)) with x1 = 1, x2 = -x^2 and x3 = -x^3. For the solutions of 2*e2 + 3*e3 = n >= 2, and the parity of e2 + e3, see the array A321201.
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EXAMPLE
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a(8) = (1/9)*[x^8] (1- x^2 - x^3)^9 = (1/9)*(-126) = -14.
a(8) = (1/9)*(- 9!/(6!*1!*2!) + 9!/(5!*0!*4!)) = -14, from the two solutions for [e2, e3], namely [1, 2] (parity odd) and [0, 4] (parity even).
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MAPLE
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f := series(x/(1 - x^2 -x^3), x, 40):
r := gfun:-seriestoseries(f, 'revogf'):
gf := convert(r, polynom) / x:
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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