A293573 -id:A293573

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A293604

Expansion of e.g.f.: exp(x * (1 - x)).

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..732

FORMULA

a(n) = (-1)^n * A000321(n).

a(n) = a(n-1) - 2 * (n-1) * a(n-2) for n > 1.

E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - Ilya Gutkovskiy, May 23 2019

a(n) = Hermite(n, 1/2). - G. C. Greubel, Jul 12 2024

MATHEMATICA

CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

PROG

(PARI) my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))

(PARI) a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017

(Magma)

R<x>:=PowerSeriesRing(Rationals(), 30);

Coefficients(R!(Laplace( Exp(x-x^2) ))); // G. C. Greubel, Jul 12 2024

(SageMath)

[hermite(n, 1/2) for n in range(31)] # G. C. Greubel, Jul 12 2024

CROSSREFS

Cf. A000321, A111884, A293571, A293572, A293573.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), this sequence (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

KEYWORD

sign

AUTHOR

Seiichi Manyama, Oct 12 2017

STATUS

approved

A293571

E.g.f.: exp(x/(1 + x + x^2)).

+10
4

1, 1, -1, -5, 25, 41, -1049, 2899, 54545, -610415, -1363409, 92652011, -651996311, -10663181255, 262674487895, -529402905149, -68312606260319, 1136414207246369, 7701376416944095, -584076369474366245, 6461047290787787321, 173442620419212050761

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Robert Israel, Table of n, a(n) for n = 0..446

Robert Israel, Plot of a(n)/(n! exp(sqrt(n))) for n = 2 .. 2500

FORMULA

E.g.f.: Product_{k>0} exp(x^(3*k-2)) / exp(x^(3*k-1)).

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). - Robert Israel, Jul 27 2020

MAPLE

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

PROG

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2))))

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)-x^(3*k-1)))))

CROSSREFS

Cf. A111884, A293572, A293573.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Oct 12 2017

STATUS

approved

A293572

E.g.f.: exp(x/(1 + x + x^2 + x^3)).

+10
4

1, 1, -1, -5, 1, 161, 31, -8021, -14335, 686881, 2925631, -91860229, -583959551, 15741408385, 169511794271, -3832934048789, -54596554106879, 1106568438159809, 23024933751472255, -412744343093399429, -11208399032299519999, 177909311974519181281

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Robert Israel, Table of n, a(n) for n = 0..446

FORMULA

E.g.f.: Product_{k>0} exp(x^(4*k-3)) / exp(x^(4*k-2)).

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) = 0. - Robert Israel, May 05 2020

MAPLE

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

PROG

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2+x^3))))

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)-x^(4*k-2)))))

CROSSREFS

Cf. A111884, A293571, A293573.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Oct 12 2017

STATUS

approved

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