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A009022 Expansion of e.g.f. cos(log(1+tanh(x))). 1
1, 0, -1, 3, -2, -20, 74, 98, -1532, 960, 41324, -105732, -1595912, 7998640, 85401224, -705417112, -6026865392, 76352075520, 537223559024, -10130428275792, -58185728893472, 1628892022801600, 7352490891960224, -313251680404802272 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A009022(x) equals (-1)^(x+1) times the imaginary part of Sum_{k=0..x-1} T(x,k)*i^k, where i is the imaginary unit. See Mathematica code below. - John M. Campbell, Nov 17 2011

LINKS

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

FORMULA

a(n) = Sum_{m=0..n/2} (-1)^(m)*Sum_{r=2*m..n} (Stirling1(r,2*m)* Sum_{k=r..n} binomial(k-1,r-1)*k!*2^(n-k)*Stirling2(n,k)*(-1)^(r+k))/r!), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 21 2011

MATHEMATICA

Join[{1}, Cos[Log[1 + Tanh[x]]];

poly[q_] := 2^(q - n)/n FunctionExpand[Sum[Binomial[n, k] k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r + 1, r + 1}]]; Table[Im[Sum[T[x, k] I^k, {k, 0, x - 1}]] (-1)^(x + 1), {x, 1, 23}]] (* John M. Campbell, Nov 17 2011 *)

With[{nn = 30}, Take[CoefficientList[Series[Cos[Log[1 + Tanh[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 1}]] (* Vincenzo Librandi, Feb 09 2014 *)

PROG

(Maxima)

a(n):=if n=0 then 1 else sum((-1)^(m)*sum((stirling1(r, 2*m)*sum(binomial(k-1, r-1)*k!*2^(n-k)*stirling2(n, k)*(-1)^(r+k), k, r, n))/r!, r, 2*m, n), m, 0, n/2); /* Vladimir Kruchinin, Jun 21 2011 */

(PARI) x='x+O('x^30); Vec(serlaplace(cos(log(1+tanh(x))))) \\ G. C. Greubel, Jul 22 2018

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cos(Log(1+Tanh(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 22 2018

CROSSREFS

Sequence in context: A065038 A123225 A009028 * A009033 A298661 A323780

Adjacent sequences:  A009019 A009020 A009021 * A009023 A009024 A009025

KEYWORD

sign,easy

AUTHOR

R. H. Hardin

EXTENSIONS

Extended with signs by Olivier Gérard Mar 15 1997

Adapted Campbell's Mathematica program for offset by Vincenzo Librandi, Feb 09 2014

STATUS

approved

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Last modified June 12 14:20 EDT 2021. Contains 344956 sequences. (Running on oeis4.)