A000831 -id:A000831 - cp's OEIS frontend

A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.

1, 1, 1, 2, 2, 2, 8, 6, 16, 40, 24, 16, 136, 240, 120, 272, 1232, 1680, 720, 272, 3968, 12096, 13440, 5040, 7936, 56320, 129024, 120960, 40320, 7936, 176896, 814080, 1491840, 1209600, 362880, 353792, 3610112, 12207360, 18627840, 13305600, 3628800

Offset: 0

N. J. A. Sloane

James Gregory calculated the first eight rows of this table (with some numerical errors) in 1671. See Roy, p. 299. - Peter Bala, Sep 06 2016

			From _Peter Bala_, Sep 06 2016: (Start)
Table begins
    1
    1     1
    2     2
    2     8      6
   16    40     24
   16   136    240    120
  272  1232   1680    720
  272  3968  12096  13440  5040
  ...
D(tan(x)) = 1 + tan(x)^2.
D^2(tan(x)) = 2*tan(x) + 2*tan(x)^3.
D^3(tan(x)) = 2 + 8*tan(x)^2 + 6*tan(x)^4.
D^4(tan(x)) = 16*tan(x) + 40*tan(x)^3 + 24*tan(x)^5. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..990
William Y. C. Chen and Amy M. Fu, The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials, arXiv:2204.01497 [math.CO], 2022.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
R. Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Mathematics Magazine Vol. 63, No. 5 (Dec., 1990), 291-306.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

Cf. A008294. Other versions of same triangle: A101343, A155100.
T(n,ceiling(n/2)) gives A000142.
Bisection of column k=0 gives A000182.
Row sums give A000831.

row[n_] := CoefficientList[ D[Tan[x], {x, n}] /. Tan -> Identity /. Sec -> Function[Sqrt[1 + #^2]], x] // DeleteCases[#, 0]&; Table[row[n], {n, 0, 10}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Apr 05 2013 *)
T[ n_, k_] := If[n<1, Boole[n==0 && k==1], (k-1)*T[n-1, k-1] + (k+1)*T[n-1, k+1]]; (* Michael Somos, Jul 08 2024 *)

{T(n, k) = if(n<1, n==0 && k==1, (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1))}; /* Michael Somos, Jul 08 2024 */

T(0, k) = delta(1, k), T(n, k) = (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1).

A258970 E.g.f.: A'(x) = 1 + A(x)^4, with A(0)=1.

Original entry on oeis.org

1, 2, 8, 80, 1088, 19328, 422912, 10987520, 330555392, 11300913152, 432717037568, 18344259092480, 852932666851328, 43157160112160768, 2360748463307620352, 138821061188696145920, 8732741520836633034752, 585172975239737913638912, 41612642758392039581155328

Offset: 0

Vaclav Kotesovec, Jun 15 2015

nonn

			A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1088*x^4/4! + 19328*x^5/5! + ...
A'(x) = 2 + 8*x + 40*x^2 + 544*x^3/3 + 2416*x^4/3 + 52864*x^5/15 + ...
1 + A(x)^4 = 2 + 8*x + 40*x^2 + 544*x^3/3 + 2416*x^4/3 + 52864*x^5/15 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A000831, A258969, A258971, A258994, A258901.

nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^4-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]

{a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^4 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

a(n) ~ n! * d^(n+1/3) / (3^(1/3) * GAMMA(1/3) * n^(2/3)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(4*j-1) = 4*sqrt(2)/(Pi + log(3-2*sqrt(2))) = 4.10260201986929...

E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^4) dx ). - Paul D. Hanna, Jun 16 2015

A258971 E.g.f.: A'(x) = 1 + A(x)^5, with A(0)=1.

Original entry on oeis.org

1, 2, 10, 130, 2330, 54770, 1591690, 55065250, 2209888250, 100922263250, 5167670934250, 293215490277250, 18260340583516250, 1238269550334211250, 90824251513716786250, 7164531681653318001250, 604824006980892825496250, 54406894886223009690031250

Offset: 0

Vaclav Kotesovec, Jun 15 2015

nonn

In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).

			A(x) = 1 + 2*x + 10*x^2/2! + 130*x^3/3! + 2330*x^4/4! + 54770*x^5/5! + ...
A'(x) = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...
1 + A(x)^5 = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258994 (k=6), A258925.

nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^5-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]

{a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^5 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

a(n) ~ n! * d^(n+1/4) / (4^(1/4) * Gamma(1/4) * n^(3/4)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(5*j-1) = 40*sqrt(5-sqrt(5)) / (8*sqrt(2)*Pi + sqrt(5+sqrt(5)) * ((9-5*sqrt(5))*log(2) + (sqrt(5)-5)*log(7+3*sqrt(5)))) = 5.53569595526739362969262739469167643400611216649309306882558956...

E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^5) dx ). - Paul D. Hanna, Jun 16 2015

A000825 Expansion of cos x (1 + sin x ) /cos 2x.

Original entry on oeis.org

1, 1, 3, 8, 57, 256, 2763, 17408, 250737, 2031616, 36581523, 362283008, 7828053417, 91620376576, 2309644635483, 31191159799808, 898621108880097, 13753735117275136, 445777636063460643, 7625476699018231808

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A000828, A000831.

Bisections are A000281 and (1/2) * A012393.

With[{nn=20},CoefficientList[Series[Cos[x] (1+Sin[x])/Cos[2x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 08 2013 *)

a(n) ~ n! * (sqrt(2) + 1 + (sqrt(2)-1)*(-1)^n) * 4^n / Pi^(n+1). - Vaclav Kotesovec, Jun 01 2015

A101343 Triangle read by rows: nonzero coefficients of the polynomials F_n(x) which express derivatives of tan(z) in terms of powers of tan(z).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 8, 2, 24, 40, 16, 120, 240, 136, 16, 720, 1680, 1232, 272, 5040, 13440, 12096, 3968, 272, 40320, 120960, 129024, 56320, 7936, 362880, 1209600, 1491840, 814080, 176896, 7936, 3628800, 13305600, 18627840, 12207360, 3610112, 353792

Offset: 0

Don Knuth, Jan 28 2005

Interpolates between factorials and tangent numbers.

			For example, D tan(z) = (tan(z))^2 + 1.
Array begins:
    1;
    1,   1;
    2,   2,
    6,   8,   2;
   24,  40,  16,
  120, 240, 136,  16;

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.

Alois P. Heinz, Rows n = 0..200, flattened
Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials, 2012. From _N. J. A. Sloane_, Oct 05 2012
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374 [math.CO], 2021.

Reflection of triangle A008293.
Column k=0 gives A000142.
Row sums give A000831.
T(2n-1,n) gives A000182 (for n>=1).

row[n_] := CoefficientList[ Derivative[n][Tan][z] /. Tan -> t /. Sec -> (Sqrt[1+t[#]^2]&), t[z]] // DeleteCases[#, 0]& // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 26 2013 *)

T(n,k):=if k=0 then Tr(n,k) else if 2*k-1=n then Tr(n,k-1) else Tr(n,k)+Tr(n,k-1);
Tr(n,i):=((sum(binomial(j+n-2*i-1,n-2*i-1)*(j+n-2*i)!*2^(2*i-j)*(-1)^(j-i)*stirling2(n,j+n-2*i),j,0,2*i))); /* Vladimir Kruchinin, May 27 2011 */

t(n,0)=n!;  t(n,k)=tr(n,k)+tr(n,k-1), k<=n/2; t(n,floor((n+1)/2)-1)=tr(n,floor((n+1)/2)-1);  tr(n,i)=((sum(j=0..2*i, binomial(j+n-2*i-1,n-2*i-1)*(j+n-2*i)!*2^(2*i-j)*(-1)^(j-i)*Stirling2(n,j+n-2*i)))). - Vladimir Kruchinin, May 27 2011
From Tom Copeland, Sep 30 2015: (Start)
Reversed rows signed and aerated are generated by [(1-x^2)D]^n x with D = d/dx, so exp(t(1-x^2)D) x = tanh(t + atanh(x)) is the e.g.f. of this reversed array (see A145271).
Reversed rows unsigned and aerated are generated by [(1+x^2)D]^n x, so exp(t(1+x^2)D) x = tan(t + atan(x)) = x + (1 +x^2)*t + (2x + 2x^3)*t^2/2! + (2 + 8x^2 + 6x^4)*t^3/3! + (16x + 40x^3 + 24x^5)*t^4/4! + ... is the e.g.f. for the matrix on p. 666 of the Knuth and Buckholtz link.
E.g.f. for this entry's aerated array 1 + (1 + x^2)*t + (2 + 2x^2)*t^2/2! + (6 + 8x^2 + 2x^4)*t^3/3! + (24 + 40^x^2 + 16x^4)*t^4/4! + ... = x * tan(t*x + atan(1/x)). (End)
From Fabián Pereyra, Apr 22 2022: (Start)
T(n,k) = (n-2k)*T(n-1,k) + (n-2k+2)*T(n-1,k-1).
E.g.f.: A(x,t) = sqrt(t)*(sqrt(t)*sin(x*sqrt(t))+cos(x*sqrt(t)))/ (sqrt(t)*cos(x*sqrt(t))-sin(x*sqrt(t))). (End)

More terms from Vladeta Jovovic and Ralf Stephan, Jan 30 2005

A000834 Expansion of exp(x)*(1 + tan(x))/(1 - tan(x)).

Original entry on oeis.org

1, 3, 9, 35, 177, 1123, 8569, 76355, 777697, 8911683, 113466729, 1589173475, 24280777617, 401898209443, 7163977596889, 136821894075395, 2787312733887937, 60331585563062403, 1382698089425999049

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..161
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).

A000834 := exp(x)*(sin(x)+cos(x))/(cos(x)-sin(x)) : for n from 0 to 200 do printf("%d %d ",n,n!*coeftayl(A000834,x=0,n)) ; end: # R. J. Mathar, Nov 19 2006
A000834 := proc(n) local i; add((-1)^(i*(i-1)/2)*4^i*binomial(n,i)*(euler(i,1/2)+euler(i,1)),i=0...n)-1 end; # Peter Luschny, Nov 25 2010

With[{nn=20},CoefficientList[Series[Exp[x] (1+Tan[x])/(1-Tan[x]), {x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Sep 08 2011 *)

a(n) = Sum_{i=0..n} binomial(n,i)*A000831(n-i). - R. J. Mathar, Nov 19 2006
a(n) := -1 + Sum_{i=0...n} ((-1)^(i(i-1)/2) 4^i C(n,i)(E_{i}(1/2) + E_{i}(1))), where E_{n}(x) are Euler polynomials. - Peter Luschny, Nov 25 2010
G.f.: G(0)*2*x/(1 - x)/(1 - 3*x) + 1/(1 - x), where G(k) = 1 - 2*x^2*(k+1)*(k+2)/(2*x^2*(k+1)*(k+2) - (2*x*k + 3*x - 1)*(2*x*k + 5*x - 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 24 2014
a(n) ~ n! * exp(Pi/4) * 2^(2*n+2) / Pi^(n+1). - Vaclav Kotesovec, Jul 02 2015

A054942 Number of connected oriented graphs on n nodes with an even number of edges.

Original entry on oeis.org

1, 0, 12, 304, 27664, 6990848, 5179182272, 11396324423680, 74944172893348096, 1476405354971703541760, 87208352627656970763963392, 15450530398306943408624578330624, 8211400756816955708062672329408385024

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000831, A054941, A054943.

m:=30;
f:= func< x | (&+[3^Binomial(n,2)*x^n/Factorial(n) : n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( Log(f(x)*(Cos(x) + Sin(x)))/2 ))); // G. C. Greubel, Apr 29 2023

nn = 15; g[z] := Sum[(1 + 2 u)^Binomial[n, 2] z^n/n!, {n, 0, nn}]; Drop[
Map[Total[#[[1 ;; Binomial[nn, 2] + 1 ;;2]]]&,Range[0,nn]!CoefficientList[
Series[Log[g[z]], {z, 0, nn}], {z, u}]], 1] (* Geoffrey Critzer, Jul 28 2016 *)

seq(n)={my(A=O(x*x^n)); Vec(serlaplace(log(sum(k=0, n, 3^binomial(k, 2)*x^k/k!) + A) + log(cos(x + A) + sin(x + A)))/2)} \\ Andrew Howroyd, Sep 10 2018

m=30
def f(x): return sum(3^binomial(n,2)*x^n/factorial(n) for n in range(m+4))
def A054941_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( log(f(x)*(cos(x) + sin(x)))/2 ).egf_to_ogf().list()
a=A054941_list(40); a[1:] # G. C. Greubel, Apr 29 2023

a(n) = (A054941(n) - (-1)^n*A000831(n-1))/2. - Andrew Howroyd, Sep 10 2018

E.g.f.: log(f(x)*(cos(x) + sin(x))), where f(x) = Sum_{j >= 0} 3^binomial(j, 2)*x^j/j!. - G. C. Greubel, Apr 29 2023

More terms from Vladeta Jovovic, Mar 11 2003

A054943 Number of connected oriented graphs on n nodes with an odd number of edges.

Original entry on oeis.org

0, 2, 8, 320, 27584, 6991360, 5179178368, 11396324458496, 74944172892993536, 1476405354971707604992, 87208352627656970712229888, 15450530398306943408625302896640, 8211400756816955708062672318337859584

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000831, A054941, A054942.

nn = 15; g[z] :=Sum[(1 + 2 u)^Binomial[n, 2] z^n/n!, {n, 0, nn}]; Drop[
Map[Total[#[[2 ;; Binomial[nn, 2] + 1 ;;2]]]&,Range[0,nn]!CoefficientList[
Series[Log[g[z]], {z, 0, nn}], {z, u}]], 1] (* Geoffrey Critzer, Jul 28 2016 *)

seq(n)={my(A=O(x*x^n)); Vec(serlaplace(log(sum(k=0, n, 3^binomial(k, 2)*x^k/k!) + A) - log(cos(x + A) + sin(x + A)))/2, -n)} \\ Andrew Howroyd, Sep 10 2018

a(n) = (A054941(n) + (-1)^n*A000831(n-1))/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 11 2003

A253165 a(n) = (-1)^n2^(6n+3)(zeta(-2n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.

Original entry on oeis.org

1, 8, 256, 17408, 2031616, 362283008, 91620376576, 31191159799808, 13753735117275136, 7625476699018231808, 5192022022552652087296, 4258996468871236847403008, 4142655008190840426050093056, 4714505177821257067736657297408, 6206008749802659037752564348092416

Offset: 0

Peter Luschny, Mar 11 2015

nonn

Cf. A000825, A000828, A000831, A012393.

a := n -> (-1)^n*2^(6*n+3)*(Zeta(0,-2*n-1,1/2)-Zeta(0,-2*n-1, 1)):
seq(a(n), n=0..14);

f[n_] := (-1)^n*2^(6 n + 3) (Zeta[-2 n - 1, 1/2] - Zeta[-2 n - 1, 1]); Array[f, 15, 0] (* Robert G. Wilson v, Mar 11 2015 *)
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/g[k+1]); gf = g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jun 01 2015, after Sergei N. Gladkovskii *)

a(n) = (-1)^n*2^(4*n+1)*(E(2*n+1,1/2)-E(2*n+1,0)), where E(n,x) are the Euler polynomials.
a(n) = A000825(2*n+1).
a(n) = A000828(2*n+1).
a(n) = A000831(2*n+1)/2.
a(n) = A012393(n+1)/2.
G.f.: S(0), where S(k)= 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/S(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 28 2015
a(n) ~ (2*n+1)! * 2^(4*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Jun 01 2015

A334317 Expansion of e.g.f. tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).

Original entry on oeis.org

1, 2, 6, 30, 198, 1638, 16254, 188190, 2490102, 37067382, 613089486, 11154460590, 221391950598, 4760331408198, 110229346777374, 2734768080189630, 72372319913943702, 2034948511063817622, 60583999401612797166, 1903897439808684195150, 62980420349165187160998

Offset: 0

Michael Somos, Apr 22 2020

nonn

If f(x) is the e.g.f. of this sequence, and if x+y+z=0, then f(x)+f(y)+f(z) = 3*f(x)*f(y)*f(z).

Seiichi Manyama, Table of n, a(n) for n = 0..415

Cf. A000831.

a[ n_] := If[n < 0, 0, n! SeriesCoefficient[Tan[Pi/3 + Sqrt[3]/2 x]/Sqrt[3], {x, 0, n}]];
With[{nn=20},CoefficientList[Series[(Tan[Pi/3+x Sqrt[3]/2])/Sqrt[3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 26 2021 *)

{a(n) = my(s=quadgen(12), A); if(n < 0, 0, A = simplify(tan(s/2*x + x*O(x^n))/s); n! * polcoeff( (1 + A)/(1 - 3*A), n))};

{a(n) = if(n<1, n==0, n<2, 2, n--; 3/2 * sum(k=0, n, binomial(n, k) * a(k) * a(n-k)))};

E.g.f.: tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).
a(n+1) = (3/2) * Sum_{k=0..n} binomial(n, k) * a(k) * a(n-k), with a(0) = 1, a(1) = 2.
a(n) ~ 2 * n! * 3^((3*n+1)/2) / Pi^(n+1). - Vaclav Kotesovec, Jul 06 2020

cp's OEIS Frontend

A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.

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A258970 E.g.f.: A'(x) = 1 + A(x)^4, with A(0)=1.

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A258971 E.g.f.: A'(x) = 1 + A(x)^5, with A(0)=1.

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A000825 Expansion of cos x (1 + sin x ) /cos 2x.

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A101343 Triangle read by rows: nonzero coefficients of the polynomials F_n(x) which express derivatives of tan(z) in terms of powers of tan(z).

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A000834 Expansion of exp(x)*(1 + tan(x))/(1 - tan(x)).

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A054942 Number of connected oriented graphs on n nodes with an even number of edges.

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SageMath

Formula

A253165 a(n) = (-1)^n2^(6n+3)(zeta(-2n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.