A317409 -id:A317409 - cp's OEIS frontend

A317406 Expansion of e.g.f. sin(x/(1 - x)).

0, 1, 2, 5, 12, 1, -450, -6931, -89096, -1120895, -14394150, -191263051, -2638282812, -37716883775, -556075744042, -8385570334051, -127637336779920, -1916072623603199, -27033275598036174, -311878728377256475, -918069644450841860, 120594465496571606401, 6362190374664242284782

Offset: 0

Ilya Gutkovskiy, Jul 27 2018

sign

Lah transform of the sequence 0, 1, 0, -1, 0, 1, 0, -1, ...

N. J. A. Sloane, Transforms

Cf. A088313, A219613, A317409.

a:=series(sin(x/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Sin[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(k + 1) Binomial[n - 1, 2 k - 2] n!/(2 k - 1)!, {k, Floor[(n + 1)/2]}], {n, 0, 22}]
Join[{0}, Table[n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1/2, 1, 3/2}, -1/4], {n, 22}]]

a(n) = Sum_{k=1..floor((n+1)/2)} (-1)^(k+1)*binomial(n-1,2*k-2)*n!/(2*k-1)!.

A317410 Expansion of e.g.f. sec(x/(1 - x)).

Original entry on oeis.org

1, 0, 1, 6, 41, 340, 3361, 38682, 508241, 7506504, 123108961, 2219822990, 43648348985, 929502984540, 21311829302401, 523455901397730, 13712375005949345, 381621247702458640, 11244620308691664961, 349715433597469496982, 11448372539225223596105, 393503844330372123056100, 14169282835811140260616801

Offset: 0

Ilya Gutkovskiy, Jul 27 2018

nonn

Lah transform of the sequence 1, 0, 1, 0, 5, 0, 61, 0, 1385, ... (A000364 interspersed with zeros).

N. J. A. Sloane, Transforms

Cf. A000364, A317409.

a:=series(sec(x/(1 - x)), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Sec[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Abs[EulerE[k]] n!/k!, {k, 0, n}], {n, 0, 22}]

x = 'x + O('x^30); Vec(serlaplace(1/cos(x/(1 - x)))) \\ Michel Marcus, Jul 28 2018

E.g.f.: sec(x/(1 - x)).

a(n) ~ 4 * n! * (Pi+2)^(n-1) / Pi^(n+1). - Vaclav Kotesovec, Jul 28 2018

cp's OEIS Frontend

A317406 Expansion of e.g.f. sin(x/(1 - x)).

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