A293604
Expansion of e.g.f.: exp(x * (1 - x)).
Original entry on oeis.org
1, 1, -1, -5, 1, 41, 31, -461, -895, 6481, 22591, -107029, -604031, 1964665, 17669471, -37341149, -567425279, 627491489, 19919950975, -2669742629, -759627879679, -652838174519, 31251532771999, 59976412450835, -1377594095061119, -4256461892701199
Offset: 0
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R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-x^2) ))); // G. C. Greubel, Jul 12 2024
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CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
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my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))
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a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017
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[hermite(n, 1/2) for n in range(31)] # G. C. Greubel, Jul 12 2024
A293571
E.g.f.: exp(x/(1 + x + x^2)).
Original entry on oeis.org
1, 1, -1, -5, 25, 41, -1049, 2899, 54545, -610415, -1363409, 92652011, -651996311, -10663181255, 262674487895, -529402905149, -68312606260319, 1136414207246369, 7701376416944095, -584076369474366245, 6461047290787787321, 173442620419212050761
Offset: 0
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f:= gfun:-rectoproc({a(n) = (3-2*n)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (5-2*n)*(n-1)*(n-2)*a(n-3)- (n-4)*(n-3)*(n-2)*(n-1)*a(n-4),a(0)=1,a(1)=1,a(2)=-1,a(3)=-5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 27 2020
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N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2))))
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N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)-x^(3*k-1)))))
A293572
E.g.f.: exp(x/(1 + x + x^2 + x^3)).
Original entry on oeis.org
1, 1, -1, -5, 1, 161, 31, -8021, -14335, 686881, 2925631, -91860229, -583959551, 15741408385, 169511794271, -3832934048789, -54596554106879, 1106568438159809, 23024933751472255, -412744343093399429, -11208399032299519999, 177909311974519181281
Offset: 0
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f:= gfun:-rectoproc(n*(n+5)*(n+4)*(n+3)*(n+2)*(n+1)*a(n) + 2*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*a(n+1) + (n+5)*(n+4)*(n+3)*(8+3*n)*a(n+2) + (n+5)*(n+4)*(13+4*n)*a(n+3) + 3*(n+4)*(n+5)*a(n+4) + (9+2*n)*a(n+5) + a(n+6),
a(0) = 1, a(1) = 1, a(2) = -1, a(3) = -5, a(4) = 1, a(5) = 161}, a(n), remember):
map(f, [$0..25]); # Robert Israel, May 05 2020
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N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2+x^3))))
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N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)-x^(4*k-2)))))
Showing 1-3 of 3 results.
