A225688 -id:A225688

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A225689

E.g.f.: sec(x)^2*tan(x)+sec(x)*tan(x)^2.

+10
6

0, 1, 2, 8, 28, 136, 662, 3968, 24568, 176896, 1326122, 11184128, 98329108, 951878656, 9596075582, 104932671488, 1192744081648, 14544442556416, 183983154281042, 2475749026562048, 34489251602450188, 507711943253426176, 7722592644581974502, 123460740095103991808, 2035778987564783402728

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

OFFSET

0,3

COMMENTS

Number of up-down max-min permutations of n elements.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, 2013.

Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

FORMULA

The e.g.f. can also be written as sin(x)*(1+sin(x))/cos(x)^3.

A225688(n)+a(n) = A000111(n+2). - corrected by Vaclav Kotesovec, May 26 2013

a(n) ~ n! * n^2*(2/Pi)^(n+3). - Vaclav Kotesovec, May 26 2013

MATHEMATICA

Table[n!*SeriesCoefficient[Sin[x]*(1+Sin[x])/Cos[x]^3, {x, 0, n}] , {n, 0, 20}] (* Vaclav Kotesovec, May 26 2013 *)

CROSSREFS

Cf. A000111, A225688.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 26 2013

STATUS

approved

A164575

a(n) = n! * [x^n] 2*(tan(x))^2*(sec(x) + tan(x)).

+10
1

0, 0, 4, 12, 56, 240, 1324, 7392, 49136, 337920, 2652244, 21660672, 196658216, 1859020800, 19192151164, 206057828352, 2385488163296, 28669154426880, 367966308562084, 4893320282898432, 68978503204900376, 1005520890400604160, 15445185289163949004, 244890632417194278912

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

FORMULA

a(n-2) = |{up-down 2nd-max-upper permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).

a(0) = 0 and a(n) = 2*A000142(n)*Sum_{i,j,k>=0, (2*i+1)+(2*j+1)+k=n} A000111(2*i+1)*A000111(2*j+1)*A000111(k)/(A000142(2*i+1)*A000142(2*j+1)*A000142(k)) for n > 0 (see Lemma 3.6 in Kobayashi).

a(2*n) = 2*A225689(2*n) (see Lemma 4.2 in Kobayashi).

a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - Vaclav Kotesovec, Aug 12 2019

MAPLE

gf := (2*sin(x)*tan(x))/(1 - sin(x)): ser := series(gf, x, 25):

seq(n!*coeff(ser, x, n), n=0..23); # Peter Luschny, Aug 19 2019

MATHEMATICA

CoefficientList[Series[2Tan[x]^2(Sec[x]+Tan[x]), {x, 0, 23}], x]*Table[n!, {n, 0, 23}]

PROG

(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(serlaplace(2*(tan(x))^2*(1/cos(x) + tan(x))))) \\ Michel Marcus, Aug 13 2019

CROSSREFS

Cf. A000111, A000142, A000182, A000364, A009764, A013525, A024283, A181937, A225688, A225689.

KEYWORD

nonn,easy

AUTHOR

Stefano Spezia, Aug 12 2019

STATUS

approved

A309845

Expansion of e.g.f.: sec(x) + 2*tan(x).

+10
0

1, 2, 1, 4, 5, 32, 61, 544, 1385, 15872, 50521, 707584, 2702765, 44736512, 199360981, 3807514624, 19391512145, 419730685952, 2404879675441, 58177770225664, 370371188237525, 9902996106248192, 69348874393137901, 2030847773013704704, 15514534163557086905, 493842960380415967232

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, 2013.

Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

FORMULA

a(n-2) = |{up-down 2nd-max-lower permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).

a(n) = A000111(n+2) - A164575(n) (See Definition 3.4 in Kobayashi).

a(n) = A225688(n) + A225689(n) - A164575(n) (See Heneghan-Petersen and Kobayashi articles).

a(2*n) = A000111(2*n) (See Lemma 3.8 in Kobayashi).

a(2*n+1) = 2*A000111(2*n+1) (See Lemma 3.8 in Kobayashi).

MATHEMATICA

CoefficientList[Series[Sec[x]+2Tan[x], {x, 0, 25}], x]*Table[n!, {n, 0, 25}]

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ Michel Marcus, Aug 20 2019

CROSSREFS

Cf. A000111, A000182, A000364, A164575, A181937, A225688, A225689.

KEYWORD

nonn,easy

AUTHOR

Stefano Spezia, Aug 20 2019

STATUS

approved

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