A003707 -id:A003707 - cp's OEIS frontend

A000816 E.g.f.: Sum_{n >= 0} a(n) * x^(2n) / (2n)! = sin(x)^2 / cos(2*x).

0, 2, 40, 1952, 177280, 25866752, 5535262720, 1633165156352, 635421069967360, 315212388819402752, 194181169538675507200, 145435130631317935357952, 130145345400688287667978240, 137139396592145493713802493952

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..200

Cf. A000364, A000819, A000822, A000828, A003707, A009125, A009569.

Union[ Range[0, 26]! CoefficientList[ Series[ Sin[x]^2/Cos[ 2x], {x, 0, 26}], x]] (* Robert G. Wilson v, Apr 16 2011 *)
Table[(-1)^(n + 1) 2^(2 n) I PolyLog[-2 n, I], {n, 1, 13}] (* Artur Jasinski, Mar 21 2022 *)
With[{nn=30},Take[CoefficientList[Series[Sin[x]^2/Cos[2x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Oct 18 2024 *)

{a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (2 - 1 / cos(x + x * O(x^m))^2) - 1, m))} /* Michael Somos, Apr 16 2011 */

@CachedFunction
def sp(n,x) :
    if n == 0 : return 1
    return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
def A000816(n) : return 0 if n == 0 else abs(sp(2*n,x)/2)
[A000816(n) for n in (0..13)]   # Peter Luschny, Jul 30 2012

(1/2) * A002436(n), n > 0. - Ralf Stephan, Mar 09 2004
a(n) = 2^(2*n - 1) * A000364(n) except at n=0.
E.g.f.: sin(x)^2/cos(2x) = 1/Q(0) - 1/2; Q(k) = 1 + 1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
a(n) = A000819(n) unless n=0.
G.f.: (1/(G(0))-1)/2 where G(k) = 1 - 4*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
G.f.: T(0)/2 - 1/2, where T(k) = 1 - 4*x*(k+1)^2/( 4*x*(k+1)^2 - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
E.g.f.: sin(x)^2/cos(2*x) = x^2/(1-2*x^2)*T(0), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/( x^2*(2*k+1)*(2*k+2) + ((k+1)*(2*k+1) - 2*x^2)*((k+2)*(2*k+3) - 2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
From Artur Jasinski, Mar 21 2022: (Start)
For n > 0:
a(n) = Pi^(2*n-1)*(-Psi(2*n,1/4) - (4^n)*(2^(2*n+1)-1)*Gamma(2*n+1)*Zeta(2*n+1)).
a(n) = (-1)^(n+1)*2^(2*n)*i*Li_(2*n,i) where i=sqrt(-1) and Li is polylogarithm function.
a(n) = (-64)^n*(zeta(-2*n,1/4)-zeta(-2*n,3/4)) where zeta is Hurwitz zeta function.
a(n) = (-16)^n*lerchphi(-1,-2*n,1/2). (End)

A354065 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tan(x).

Original entry on oeis.org

1, -2, 2, -8, 56, -496, 3184, -22784, 273920, -4539136, 48104704, -506000384, 10591523840, -204528633856, 2888557717504, -53417657237504, 1249919350046720, -28453501844586496, 624022403933077504, -13729309300086800384, 372737701735949926400, -11010228423219933085696

Offset: 1

Ilya Gutkovskiy, May 16 2022

sign

Cf. A000182, A003707, A009006, A353583, A353584, A353611, A353911, A354055, A354056, A354063, A354064, A354066.

nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

E.g.f.: Sum_{k>=1} mu(k) * log(1 + tan(x^k)) / k.

A024299 a(n) = (2n)! [x^(2n)] log(1 + tanh(x)^2)/2.

Original entry on oeis.org

0, 1, -14, 496, -34544, 4055296, -724212224, 183218384896, -62380415842304, 27507260369207296, -15250924309151350784, 10384039093607251050496, -8517991922318587187953664, 8285309769460200661892202496, -9429010285390912531529354706944

Offset: 0

R. H. Hardin

sign

G. C. Greubel, Table of n, a(n) for n = 0..215

Cf. A009403.

Cf. A003707, A101921.

L:=RiemannZeta();
[0] cat [-Round(2^(2*n-1)*(4^n-2)*(4^n-1)*Evaluate(L, 1-2*n)): n in [1..15]]; // G. C. Greubel, Jul 12 2022

a := n -> `if`(n=0, 0, -2^(2*n-1)*(4^n-2)*(4^n-1)*Zeta(1-2*n)):
seq(a(n), n=0..14); # Peter Luschny, Oct 29 2020

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]^2]/2,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Dec 12 2021 *)

my(x='x+O('x^30), v = concat([0,0], Vec(serlaplace (log(1+tanh(x)^2)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Oct 29 2020

[0]+[-2^(2*n-1)*(4^n-2)*(4^n-1)*zeta(1-2*n) for n in (1..15)] # G. C. Greubel, Jul 12 2022

a(n) = A009403(n)/2.

a(n) = -2^(2*n-1)*(4^n - 2)*(4^n - 1)*zeta(1-2*n) for n >= 1. - Peter Luschny, Oct 29 2020

Extended with signs, Mar 1997

Previous Mathematica program replaced by Harvey P. Dale, Dec 12 2021

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.

a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))

{a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */

def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.

G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177

Offset: 1

Johannes W. Meijer, Jul 27 2016

sign

The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 1..451
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.

a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)

N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).

E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 777, 7379, 80983, 1007137, 13986289, 214383171, 3593224767, 65347120705, 1281151315641, 26928292883795, 603928982033863, 14392387319349697, 363135896514611041, 9669298448057196291, 270932711729869233903, 7967970654277850949025

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..427

Cf. A000111, A000828, A003707, A004211, A006229, A331607.

S:= series(exp(1/(1-tan(x))-1), x, 31):
seq(coeff(S,x,i)*i!, i=0..30); # Robert Israel, Dec 10 2024

nmax = 20; CoefficientList[Series[Exp[1/(1 - Tan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] 2^(k - 1) A000111[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(sin(x) / (cos(x) - sin(x))).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * 2^(k-1) * A000111(k) * a(n-k).
a(n) ~ 2^(2*n - 1/4) * exp(1/Pi - 1/2 + 2^(3/2)*sqrt(n/Pi) - n) * n^(n - 1/4) / Pi^(n + 1/4). - Vaclav Kotesovec, Jan 27 2020

A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -2, 18, -312, 9470, -436860, 28616322, -2522596496, 288046961190, -41355026494020, 7291524732108650, -1548849359704927896, 390122366308850972238, -114968364853645904762252, 39189956630839558368115410, -15300235972710835734174638880

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296838.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tan(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A331978 E.g.f.: -log(2 - cosh(x)) (even powers only).

Original entry on oeis.org

0, 1, 4, 46, 1114, 46246, 2933074, 263817646, 31943268634, 5009616448246, 987840438629794, 239217148602642046, 69790939492563608554, 24143849395162438623046, 9772368696995766705116914, 4575221153658910691872135246, 2453303387149157947685779986874

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

Cf. A000111, A000182, A000629, A003703, A003704, A003707, A094088, A331611.

ptan := proc(n) option remember;
    if irem(n, 2) = 0 then 0 else
    add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1, 2) fi end:
A331978 := n -> ptan(2*n - 1):
seq(A331978(n), n = 0..16);  # Peter Luschny, Jun 06 2022

nmax = 16; Table[(CoefficientList[Series[-Log[2 - Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(0) = 0; a(n) = A094088(n) - (1/n) * Sum_{k=1..n-1} binomial(2*n,2*k) * A094088(n-k) * k * a(k).

a(n) ~ (2*n)! / (n * log(2 + sqrt(3))^(2*n)). - Vaclav Kotesovec, Feb 07 2020

A107729 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k < 0 or if k > 0; T(n,k) = kT(n-1,k-1) + (k+2)T(n-1,k+1).

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 8, 0, 6, 16, 0, 40, 0, 24, 0, 136, 0, 240, 0, 120, 272, 0, 1232, 0, 1680, 0, 720, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880, 353792, 0

Offset: 0

N. J. A. Sloane, Jun 10 2005

Triangle is related to the tangent numbers A000182.

			Triangle begins:
     1;
     0,    1;
     2,    0,    2;
     0,    8,    0,    6;
    16,    0,   40,    0,    24;
     0,   136,   0,   240,    0,   120;
    272,   0,  1232,   0,   1680,   0,    720;
     0,  3968,   0,  12096,   0,  13440,   0,  5040;
   7936,   0,  56320,  0,  129024,  0,  120960,  0,   40320;
     0, 176896,  0, 814080,   0, 1491840,  0, 1209600,  0, 362880;
  353792,  0, 3610112, 0, 12207360, 0, 18627840, 0, 13305660, 0, 3628800;
  ...

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 446.

Similar to A104035. Leading edge is essentially A000182.

Cf. A003707.

T:=proc(n,k) if k=-1 then 0 elif n=1 and k=1 then 1 elif k>n then 0 else (k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form [Produces triangle with a different offset] # Emeric Deutsch, Jun 13 2005

T(n, n) = n!; T(n, 0) = 0 if n = 2m+1; T(n, 0) = A000182(m+1) if n = 2m.
Sum_{k>=0} T(m, k)*T(n, k)*(k+1) = T(m+n, 0).
Sum_{k>=0} T(n, k) = |A003707(n+1)|.

More terms from Emeric Deutsch, Jun 13 2005
Additional comments from Philippe Deléham, Sep 17 2005
Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A000816 E.g.f.: Sum_{n >= 0} a(n) * x^(2*n) / (2*n)! = sin(x)^2 / cos(2*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A354065 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tan(x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A024299 a(n) = (2*n)! [x^(2*n)] log(1 + tanh(x)^2)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

A274805 The logarithmic transform of sigma(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A000816 E.g.f.: Sum_{n >= 0} a(n) * x^(2n) / (2n)! = sin(x)^2 / cos(2*x).

A024299 a(n) = (2n)! [x^(2n)] log(1 + tanh(x)^2)/2.

A107729 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k < 0 or if k > 0; T(n,k) = kT(n-1,k-1) + (k+2)T(n-1,k+1).