A024299 -id:A024299 - cp's OEIS frontend

A101921 a(2n) = a(n) + 2n - 1, a(2n+1) = 4n.

0, 1, 4, 4, 8, 9, 12, 11, 16, 17, 20, 20, 24, 25, 28, 26, 32, 33, 36, 36, 40, 41, 44, 43, 48, 49, 52, 52, 56, 57, 60, 57, 64, 65, 68, 68, 72, 73, 76, 75, 80, 81, 84, 84, 88, 89, 92, 90, 96, 97, 100, 100, 104, 105, 108, 107, 112, 113, 116, 116, 120, 121, 124, 120, 128

Offset: 1

Ralf Stephan, Dec 21 2004

nonn

Exponent of 2 in tangent numbers A000182.
Also, exponent of 2 in the sequences A008775, A009670, A009764, A009798, A012227, A024236, A024277, A024299, A052510.
Also, exponent of 2 in 4^(n-1)/n. [David Brink, Aug 08 2013]

			G.f. = x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 9*x^6 + 12*x^7 + 11*x^8 + 16*x^9 + 17*x^10 + ...

Iain Fox, Table of n, a(n) for n = 1..10000

Cf. A000182, A007814.

Cf. A008775, A009670, A009764, A009798, A012227, A024236, A024277, A024299, A052510.

a[n_]:= If[n<1, 0, 2n -2 - IntegerExponent[n, 2]]; (* Michael Somos, Mar 02 2014 *)

a(n)=valuation(4^(n-1)/n,2); \\ Joerg Arndt, Aug 13 2013

def A101921(n): return (n-1<<1)-(~n & n-1).bit_length() # Chai Wah Wu, Apr 14 2023

[2*n-2 -valuation(n,2) for n in (1..100)] # G. C. Greubel, Nov 29 2021

a(n) = 2n - 2 - A007814(n).
a(n) = A007814(A000182(n)).
G.f.: Sum_{k>=0} t^2*(1+4*t+t^2)/(1-t^2)^2 where t=x^2^k.

A003707 Expansion of e.g.f. log(1 + tan(x)).

Original entry on oeis.org

0, 1, -1, 4, -14, 80, -496, 3904, -34544, 354560, -4055296, 51733504, -724212224, 11070525440, -183218384896, 3266330312704, -62380415842304, 1270842139934720, -27507260369207296, 630424777638805504, -15250924309151350784, 388362339077351014400, -10384039093607251050496

Offset: 0

R. H. Hardin, Simon Plouffe

sign

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

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.

Bisections are A002436 and |A024299|.

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

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

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

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 */

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 */

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

def A003707_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( log(1 +tan(x)) ).egf_to_ogf().list()
A003707_list(25) # G. C. Greubel, Jun 08 2020

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

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

A009403 Expansion of e.g.f. log(1 + tanh(x)^2), even powers only.

Original entry on oeis.org

0, 2, -28, 992, -69088, 8110592, -1448424448, 366436769792, -124760831684608, 55014520738414592, -30501848618302701568, 20768078187214502100992, -17035983844637174375907328, 16570619538920401323784404992

Offset: 0

R. H. Hardin

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A024299.

L:=RiemannZeta();
[0] cat [-Round(4^n*(4^n-2)*(4^n-1)*Evaluate(L, 1-2*n)): n in [1..20]]; // G. C. Greubel, Jul 12 2022

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]^2],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 27 2013 *)

[0]+[-4^n*(4^n-2)*(4^n-1)*zeta(1-2*n) for n in (1..20)] # G. C. Greubel, Jul 12 2022

a(n) ~ (-1)^(n+1) * 2^(4*n) * (2*n)! / (n * Pi^(2*n)). - Vaclav Kotesovec, Apr 20 2014
From G. C. Greubel, Jul 12 2022: (Start)
a(n) = 2*A024299(n).
a(n) = -4^n * (4^n - 2)*(4^n - 1)*Zeta(1-2*n), with a(0) = 0. (End)

Extended with signs by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Aug 27 2013

cp's OEIS Frontend

A101921 a(2n) = a(n) + 2n - 1, a(2n+1) = 4n.

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A003707 Expansion of e.g.f. log(1 + tan(x)).

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A009403 Expansion of e.g.f. log(1 + tanh(x)^2), even powers only.

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Magma

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