cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A225689 E.g.f.: sec(x)^2*tan(x)+sec(x)*tan(x)^2.

Original entry on oeis.org

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
Offset: 0

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Author

N. J. A. Sloane, May 26 2013

Keywords

Comments

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

Links

Crossrefs

Cf. A000111, A225688.

Programs

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

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

A164575 a(n) = n! * [x^n] 2*(tan(x))^2*(sec(x) + tan(x)).

Original entry on oeis.org

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

Views

Author

Stefano Spezia, Aug 12 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]
  • 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

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

A309845 Expansion of e.g.f.: sec(x) + 2*tan(x).

Original entry on oeis.org

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
Offset: 0

Views

Author

Stefano Spezia, Aug 20 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    CoefficientList[Series[Sec[x]+2Tan[x],{x,0,25}],x]*Table[n!,{n,0,25}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ Michel Marcus, Aug 20 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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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