A009611 -id:A009611 - cp's OEIS frontend

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

0, 1, 1, 18, 227, 4565, 126648, 4620805, 213569269, 12165013026, 835868220455, 68093897815361, 6483538063860336, 712877916658802713, 89586864207214060057, 12753583150716684461970, 2040805972702652020364603, 364567588100855831300341565

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			sinh(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + 4565*x^10/10! + ...

Cf. A001469, A003717, A009611, A110501, A296839, A296841, A296842, A296853, A296856.

nmax = 17; Table[(CoefficientList[Series[Sinh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] sinh(x*tan(x/2)).

A024263 Expansion of sinh(tan(x)*x)/2.

Original entry on oeis.org

0, 1, 4, 108, 4448, 276560, 24034752, 2782112704, 413640580096, 76768625740032, 17380983003345920, 4711153236657376256, 1505098422997799215104, 559357234754573189828608

Offset: 0

R. H. Hardin

nonn

Limit n->infinity A024263(n)/A009252(n) = 1/4. - Vaclav Kotesovec, Jan 20 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..238

Cf. A009252, A009611.

Sinh[ Tan[ x ]*x ]/2 (* Even Part *)
With[{nn=40}, Take[CoefficientList[Series[Sinh[Tan[x]*x]/2, {x, 0, nn}], x] * Range[0, nn]!, {1, -1, 2}]] (* Vaclav Kotesovec, Jan 20 2015 *)

a(n) ~ n^(2*n-1/4) * 2^(4*n-7/4) * exp(2*sqrt(2*n)-2*n-1/2) / Pi^(2*n) * (1 - (Pi^2-1)/(12*sqrt(2*n))). - Vaclav Kotesovec, Jan 20 2015

Extended and signs tested 03/97.

cp's OEIS Frontend

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

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