A003718 -id:A003718 - cp's OEIS frontend

A342604 a(n) = Sum_{j=1..n} A003718(j-1)*prime(j).

2, 5, 10, 17, 39, 52, 69, 126, 195, 224, 255, 403, 649, 821, 868, 921, 1216, 1826, 2496, 2851, 2924, 3003, 3501, 4836, 6776, 8291, 8909, 9016, 9125, 9916, 12583, 17168, 21963, 24882, 25925, 26076, 26233, 27537, 32213, 41901, 54431, 64567, 69915, 71459, 71656, 71855, 73754, 81782, 100850, 129704

Offset: 1

J. M. Bergot and Robert Israel, Mar 16 2021

nonn

			a(3) = A003718(0)*prime(1) + A003718(1)*prime(2) + A003718(2)*prime(3) = 1*2 + 1*3 + 1*5 = 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003718, A342605, A342606.

p:= 1: R:= NULL:
for n from 0 to 14 do
  for k from 0 to n do
    p:= nextprime(p);
    R:= R, binomial(n,k)*p
od od:
ListTools:-PartialSums([R]):

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...

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

Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.

nmax = 16; Table[(CoefficientList[Series[Tan[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)] tan(x*tan(x/2)).

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - Vaclav Kotesovec, Dec 21 2017, updated Mar 16 2024

A003721 Expansion of e.g.f. tan(tanh(x)) (odd powers only).

Original entry on oeis.org

1, 0, -8, 112, -128, -109824, 8141824, -353878016, -9666461696, 5151942574080, -825073851170816, 76429076694827008, 2051308253366714368, -2361338488910424047616, 171581865952588387581952

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).

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A003716, A003718.

With[{nn=30},Take[CoefficientList[Series[Tan[Tanh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Nov 05 2011 *)

a(n):=sum(((sum(j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*stirling2(2*i+1,j),j,1,2*i+1))*sum(binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*stirling2(2*n-1,k),k,2*i+1,2*n-1))/(2*i+1)!,i,0,(n-1)); /* Vladimir Kruchinin, Jun 10 2011 */

a(n) = Sum_{i=0..(n-1)} ( ( Sum_{j=1..2*i+1} j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*Stirling2(2*i+1,j) ) * Sum_{k=2*i+1..2*n-1} binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*Stirling2(2*n-1,k) )/(2*i+1)!. - Vladimir Kruchinin, Jun 10 2011

A296465 Expansion of e.g.f. arctanh(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 4, 88, 4688, 459520, 71876352, 16428530688, 5167215464448, 2140879726411776, 1130276555155243008, 740796870212763254784, 590192778209307913617408, 561748717440430309770264576, 629564244208933873601143111680, 820602153197407426121272991416320, 1230877720962045060728502509025361920

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arctanh(arctanh(x)) = x/1! + 4*x^3/3! + 88*x^5/5! + 4688*x^7/7! + 459520*x^9/9! + ...

Cf. A003718, A010050, A012263, A296467.

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

E.g.f.: arctan(arctan(x)) (odd powers only, absolute values).
E.g.f.: (log(2 - log(1 - x) + log(1 + x)) - log(2 + log(1 - x) - log(1 + x)))/2 (odd powers only).
a(n) ~ (2*n)! * ((exp(2) + 1)/(exp(2) - 1))^(2*n+1). - Vaclav Kotesovec, Dec 13 2017

A009707 Expansion of e.g.f. tan(tan(x)*x) (even powers only).

Original entry on oeis.org

0, 2, 8, 336, 15616, 1450240, 185032704, 33566984192, 7971973332992, 2424984197529600, 915532582868746240, 420569934453637906432, 230845747512083447021568, 149228982402223336708898816

Offset: 0

R. H. Hardin

nonn

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

Cf. A000182, A003718, A009752, A024265.

nmax = 20; Table[(CoefficientList[Series[Tan[x*Tan[x]], {x, 0, 2*nmax}], x] * Range[0, 2 nmax]!)[[n]], {n, 1, 2*nmax + 1, 2}] (* Vaclav Kotesovec, Dec 21 2017 *)

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 3.9786913954409425781217887822690623430980810... is the root of the equation tan(2/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.4057183994645... - Vaclav Kotesovec, Dec 21 2017

Extended and signs tested Mar 15 1997 by Olivier Gérard.

A003716 Expansion of e.g.f. tan(sinh(x)) (odd powers only).

Original entry on oeis.org

1, 3, 37, 1015, 47881, 3459819, 354711853, 48961863007, 8754050024209, 1967989239505875, 543326939019354421, 180718022989699819207, 71275877445849484090393, 32890432371345908634652347, 17555593768891213894861569085

Offset: 0

R. H. Hardin

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A003718, A003721.

Tan[ Sinh[ x ] ] (* Odd Part *)
nn = 20; Table[(CoefficientList[Series[Tan[Sinh[x]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)

a(n):=sum(((-1)^(k-1)+1)/(2^k*k!)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)*(sum(j!*2^(k-j-1)*(-1)^((k+1)/2+j)*stirling2(k,j),j,1,k)),k,1,n); /* Vladimir Kruchinin, Apr 20 2011 */

a(n) = Sum_{k=1..n} ((-1)^(k-1)+1)/(2^k*k!) * ( Sum_{i=0..k} (-1)^i*(k-2*i)^n *binomial(k,i) ) * ( Sum_{j=1..k} j! * 2^(k-j-1) * (-1)^((k+1)/2+j) * stirling2(k,j) ). - Vladimir Kruchinin, Apr 20 2011

a(n) ~ 4 * (2*n+1)! / (sqrt(4+Pi^2) * (log((Pi + sqrt(4+Pi^2))/2))^(2*n+2)). - Vaclav Kotesovec, Feb 16 2015

A009817 Expansion of e.g.f. tanh(tan(x)*x) (even powers only).

Original entry on oeis.org

0, 2, 8, -144, -11264, -323840, 36347904, 8388294656, 620426657792, -168731721990144, -79627879039631360, -11099857853570613248, 5627175549005859913728, 4469405304978356270268416

Offset: 0

R. H. Hardin, Mar 15 1996

sign

Iain Fox, Table of n, a(n) for n = 0..231 (first 101 terms from Vincenzo Librandi)

Cf. A000182, A003718, A009707, A024257.

S:= series(tanh(tan(x)*x),x,41):
seq(coeff(S,x,n)*n!,n=0..40,2); # Robert Israel, Dec 21 2017

With[{nn=30},Take[CoefficientList[Series[Tanh[Tan[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Oct 28 2013 *)

first(n) = x='x+O('x^(2*n-1)); vecextract(Vec(serlaplace(tanh(tan(x)*x)), -2*n), (4^n - 1)/3) \\ Iain Fox, Dec 21 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Oct 28 2013

A296790 Expansion of e.g.f. sec(x*sec(x)) (even powers only).

Original entry on oeis.org

1, 1, 17, 601, 38849, 4022641, 609933521, 127391254537, 35067716300033, 12304447787106529, 5360597104269331985, 2839145693984474128057, 1796556232541725248396737, 1338623568393194541863879761, 1160057210771530210422755155409, 1156898060700987368136296212581481

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

nonn

			sec(x*sec(x)) = 1 + x^2/2! + 17*x^4/4! + 601*x^6/6! + 38849*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Cf. A000364, A003712, A003718, A009008, A009009, A009010, A009011, A009015, A009118, A009562, A009765, A102072, A102075, A296731, A296740, A296791.

nmax = 15; Table[(CoefficientList[Series[Sec[x Sec[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] sec(x*sec(x)).

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 4.5851486299312178337601256220116584724159... is the real root of the equation sqrt(d) * cos(2/sqrt(d)) = 4/Pi and c = 1.99453594228967461336... - Vaclav Kotesovec, Dec 21 2017

A013519 Numerator of [x^(2n+1)] in the Taylor expansion tan(cosec(x)-cot(x)).

Original entry on oeis.org

1, 1, 3, 181, 59, 3455, 3332389, 173339393, 449509681, 198232237033, 1032475444411, 101635950910031, 71432133543444211, 671713726985602398481

Offset: 0

Patrick Demichel (patric.demichel(AT)hp.com)

The e.g.f. of x/2, tan(cosec(x)-cot(x)) = x/(2^1*1!) + 4*x^3/(2^3*3!) +72*x^5/(2^5*5!) + 2896*x^7/(2^7*7!) +.. is apparently covered by A003718.

			x/2 +x^3/12 +3*x^5/160 +181*x^7/40320 +59*x^9/53760 +3455*x^11/12773376 +...

G. C. Greubel, Table of n, a(n) for n = 0..125

Numerator[Take[CoefficientList[Series[Tan[Csc[x] - Cot[x]], {x, 0, 25}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Name edited by R. J. Mathar, Dec 19 2011

A228841 E.g.f.: sec(sec(x)-1) (even-indexed coefficients only).

Original entry on oeis.org

1, 0, 3, 75, 3108, 205125, 19839633, 2643131400, 463873573803, 103710628476075, 28775903316814668, 9702563010998171325, 3907429085273025561153, 1852516229654506870381200, 1021325008815288529961197683, 647900078249178232882473232875

Offset: 0

Geoffrey Critzer, Nov 10 2013

nonn

Call a zig permutation a permutation p(1),p(2),...,p(2n) such that p(1)>p(2)< ... > p(2n) Cf. A000364. Consider the set of all set partitions of {1,2,...,2n} into an even number of even sized blocks. a(n) is the number of ways to build a zig permutation on each block and then build a zig permutation on the set formed from a representative (say the smallest element) of each block.

			a(3) = 75.  There are 15 set partitions of {1,2,3,4,5,6} that have an even number of even sized blocks Cf. A059386.  They all have the same structure: 2/4.  We build a zig permutation on each block in 1*5=5 ways.  For each of these we then build a  zig permutation on a representative from each of the 2 blocks in 1 way.  So 5*1=5 and there are 15 such partitions so 5 *15 =75.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A219613, A080832, A003718.

nn=30;Insert[Select[Range[0,nn]!CoefficientList[Series[Sec[Sec[x]-1],{x,0,nn}],x],#>0&],0,2]

cp's OEIS Frontend

A342604 a(n) = Sum_{j=1..n} A003718(j-1)*prime(j).

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Maple

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

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Mathematica

Formula

A003721 Expansion of e.g.f. tan(tanh(x)) (odd powers only).

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Mathematica

Maxima

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A296465 Expansion of e.g.f. arctanh(arctanh(x)) (odd powers only).

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Mathematica

Formula

A009707 Expansion of e.g.f. tan(tan(x)*x) (even powers only).

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Mathematica

Formula

Extensions

A003716 Expansion of e.g.f. tan(sinh(x)) (odd powers only).

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Mathematica

Maxima

Formula

A009817 Expansion of e.g.f. tanh(tan(x)*x) (even powers only).

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Maple

Mathematica

PARI

Extensions

A296790 Expansion of e.g.f. sec(x*sec(x)) (even powers only).

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