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A003707 Expansion of e.g.f. log(1 + tan(x)).
(Formerly M3490)
10
0, 1, -1, 4, -14, 80, -496, 3904, -34544, 354560, -4055296, 51733504, -724212224, 11070525440, -183218384896, 3266330312704, -62380415842304, 1270842139934720, -27507260369207296, 630424777638805504, -15250924309151350784, 388362339077351014400, -10384039093607251050496 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

FORMULA

a(n) = Sum_{k=1..n} (-1)^(k+1) * evenp(n+k) * (-1)^((n+k)/2)/k * Sum_{j=k..n} j! * Stirling2(n, j) * 2^(n-j) * (-1)^(n+j-k) * binomial(j-1,k-1). [Vladimir Kruchinin, Aug 18 2010] [Corrected by Petros Hadjicostas, Jun 05 2020]

a(n) = Sum_{m=0..(n-1)/2} Sum_{j=0..2*m} binomial(j+n-2*m-1, n-2*m-1) * (j+n-2*m)! * 2^(2*m-j) * (-1)^(n-m+j-1) * Stirling2(n, j+n-2*m)/(n-2*m). [Vladimir Kruchinin, Jan 21 2012]

a(n) ~ (-1)^(n+1) * 4^n * (n-1)! / Pi^n. - Vaclav Kotesovec, Feb 16 2015

MAPLE

seq(coeff(series( log(1 +tan(x)), x, n+1)*n!, x, n), n = 0..25); # G. C. Greubel, Jun 08 2020

MATHEMATICA

With[{nn = 30}, CoefficientList[Series[Log[1 + Tan[x]], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Apr 11 2014 *)

PROG

(Maxima) a(n):=sum((-1)^(k+1)*if evenp(n+k) then (-1)^((n+k)/2)/k*sum(j!*stirling2(n, j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n) else 0, k, 1, n);  /* Vladimir Kruchinin, Aug 18 2010 */ /* Corrected by Petros Hadjicostas, Jun 05 2020 */

(Maxima) a(n):=sum(sum(binomial(j+n-2*m-1, n-2*m-1)*(j+n-2*m)!*2^(2*m-j)*(-1)^(n-m+j-1)*stirling2(n, j+n-2*m), j, 0, 2*m)/(n-2*m), m, 0, (n-1)/2); /* Vladimir Kruchinin, Jan 21 2012 */

(PARI) my(x='x+O('x^66)); concat([0], Vec(serlaplace(log(1+tan(x))))) \\ Joerg Arndt, Sep 02 2013

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 25); [0] cat Coefficients(R!(Laplace( Log(1 + Tan(x)) ))); // G. C. Greubel, Jun 08 2020

(Sage)

def A003707_list(prec):

    P.<x> = PowerSeriesRing(QQ, prec)

    return P( log(1 +tan(x)) ).egf_to_ogf().list()

A003707_list(25) # G. C. Greubel, Jun 08 2020

CROSSREFS

Bisections are A002436 and |A024299|.

Sequence in context: A186638 A187847 A277039 * A063862 A222497 A327355

Adjacent sequences:  A003704 A003705 A003706 * A003708 A003709 A003710

KEYWORD

sign

AUTHOR

R. H. Hardin, Simon Plouffe

EXTENSIONS

Name corrected, more terms, Joerg Arndt, Sep 02 2013

STATUS

approved

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Last modified June 15 08:14 EDT 2021. Contains 345048 sequences. (Running on oeis4.)