cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A320958 The exponential limit of arcsin (odd indices only).

Original entry on oeis.org

1, 5, 468, 197325, 233145675, 605979974250, 2987147975582925, 25254853526009732625, 340477692051264295027500, 6926101229658271208893970625, 203562520854789108487169894574375, 8346651541805126492397454664310896250, 463877742240727904202821053051014479795625
Offset: 0

Views

Author

Peter Luschny, Nov 08 2018

Keywords

Comments

See A320956 for definitions and comments.

Examples

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0, 0, 0,   0, 0,      0, 0,         0, ...
   [1] 0, 1, 0, 1, 0,   9, 0,    225, 0,     11025, ... A177145, A001818
   [2] 0, 1, 0, 4, 0, 144, 0,  14400, 0,   2822400, ... A122747
   [3] 0, 1, 0, 5, 0, 369, 0,  82125, 0,  36173025, ...
   [4] 0, 1, 0, 5, 0, 459, 0, 160875, 0, 121837275, ...
   [5] 0, 1, 0, 5, 0, 468, 0, 192375, 0, 198472050, ...
   [6] 0, 1, 0, 5, 0, 468, 0, 197100, 0, 227644200, ...
   [7] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 232737750, ...
   [8] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233134650, ...
   [9] 0, 1, 0, 5, 0, 468, 0, 197325, 0, 233145675, ...

Crossrefs

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), this sequence (arcsin),
A320959 (arctanh).
Cf. A177145, A001818, A122747.

Programs

  • Maple
    # Function ExpLim defined in A320956.
L := [ExpLim(28, arcsin)]: seq(L[2*n], n=1..13);
  • Mathematica
    m = 13; CoefficientList[ArcSin[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, 1, 0, 1, 8, 0, 1, 88, 136, 0, 1, 816, 6240, 3968, 0, 1, 7376, 195216, 513536, 176896, 0, 1, 66424, 5352544, 39572864, 51880064, 11184128, 0, 1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0, 1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0, 1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0
Offset: 0

Views

Author

Paul D. Hanna, Dec 14 2018

Keywords

Comments

Equals a row reversal of triangle A325222.
Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).
Compare also to Michael Pawellek's generalized elliptic functions.

Examples

			E.g.f.: C(x,k) = 1 + x^2/2! + (8*k^2 + 1)*x^4/4! + (136*k^4 + 88*k^2 + 1)*x^6/6! + (3968*k^6 + 6240*k^4 + 816*k^2 + 1)*x^8/8! + (176896*k^8 + 513536*k^6 + 195216*k^4 + 7376*k^2 + 1)*x^10/10! + (11184128*k^10 + 51880064*k^8 + 39572864*k^6 + 5352544*k^4 + 66424*k^2 + 1)*x^12/12! + (951878656*k^12 + 6453433344*k^10 + 8258202240*k^8 + 2458228480*k^6 + 139127640*k^4 + 597864*k^2 + 1)*x^14/14! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
1;
1, 0;
1, 8, 0;
1, 88, 136, 0;
1, 816, 6240, 3968, 0;
1, 7376, 195216, 513536, 176896, 0;
1, 66424, 5352544, 39572864, 51880064, 11184128, 0;
1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0;
1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0;
1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
S(x,k) = x + (2*k^2 + 1)*x^3/3! + (16*k^4 + 28*k^2 + 1)*x^5/5! + (272*k^6 + 1032*k^4 + 270*k^2 + 1)*x^7/7! + (7936*k^8 + 52736*k^6 + 36096*k^4 + 2456*k^2 + 1)*x^9/9! + (353792*k^10 + 3646208*k^8 + 4766048*k^6 + 1035088*k^4 + 22138*k^2 + 1)*x^11/11! + (22368256*k^12 + 330545664*k^10 + 704357760*k^8 + 319830400*k^6 + 27426960*k^4 + 199284*k^2 + 1)*x^13/13! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (5*k^4 + 4*k^2)*x^4/4! + (61*k^6 + 148*k^4 + 16*k^2)*x^6/6! + (1385*k^8 + 6744*k^6 + 2832*k^4 + 64*k^2)*x^8/8! + (50521*k^10 + 410456*k^8 + 383856*k^6 + 47936*k^4 + 256*k^2)*x^10/10! + (2702765*k^12 + 32947964*k^10 + 54480944*k^8 + 17142784*k^6 + 780544*k^4 + 1024*k^2)*x^12/12! + (199360981*k^14 + 3402510924*k^12 + 8760740640*k^10 + 5199585280*k^8 + 686711040*k^6 + 12555264*k^4 + 4096*k^2)*x^14/14! + ...

Links

Crossrefs

Cf. A322230 (S), A322232 (D), A001818 (row sums), A002105.
Cf. A325222 (row reversal).

Programs

  • PARI
    N=10;
{S=x;C=1;D=1; for(i=1,2*N, S = intformal(C*D^2 +O(x^(2*N+1))); C = 1 + intformal(S*D^2); D = 1 + k^2*intformal(S*C*D));}
for(n=0,N, for(j=0,n, print1( (2*n)!*polcoeff(polcoeff(C,2*n,x),2*j,k),", ")) ;print(""))

Formula

E.g.f. C = C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n) * k^(2*j) / (2*n)!, along with related series S = S(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C*D^2 dx.
(1b) C = 1 + Integral S*D^2 dx.
(1c) D = 1 + k^2 * Integral S*C*D dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral D^2 dx ).
(3b) D + k*S = exp( k * Integral C*D dx ).
(4a) S = sinh( Integral D^2 dx ).
(4b) S = sinh( k * Integral C*D dx ) / k.
(4c) C = cosh( Integral D^2 dx ).
(4d) D = cosh( k * Integral C*D dx ).
(5a) d/dx S = C*D^2.
(5b) d/dx C = S*D^2.
(5c) d/dx D = k^2 * S*C*D.
From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral D dx, k),
(6b) C = cn( i * Integral D dx, k),
(6c) D = dn( i * Integral D dx, k).
(7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
(7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
(7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Diagonal T(n+1,n) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.

A322232 E.g.f.: D(x,k) = 1 + k^2 * Integral S(x,k)*C(x,k)*D(x,k) dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, 0, 1, 0, 4, 5, 0, 16, 148, 61, 0, 64, 2832, 6744, 1385, 0, 256, 47936, 383856, 410456, 50521, 0, 1024, 780544, 17142784, 54480944, 32947964, 2702765, 0, 4096, 12555264, 686711040, 5199585280, 8760740640, 3402510924, 199360981, 0, 16384, 201199616, 26090711040, 419867864320, 1569971730560, 1632067372896, 441239943664, 19391512145, 0, 65536, 3220652032, 965223559168, 30892394850304, 227204970315520, 502094919789184, 353538702361888, 70347660061552, 2404879675441
Offset: 0

Views

Author

Paul D. Hanna, Dec 14 2018

Keywords

Comments

Equals a row reversal of triangle A325221.
Compare to dn(x,k) = 1 - k^2 * Integral sn(x,k)*cn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions.
Compare also to Michael Pawellek's generalized elliptic functions.

Examples

			E.g.f.: D(x,k) = 1 + k^2*x^2/2! + (5*k^4 + 4*k^2)*x^4/4! + (61*k^6 + 148*k^4 + 16*k^2)*x^6/6! + (1385*k^8 + 6744*k^6 + 2832*k^4 + 64*k^2)*x^8/8! + (50521*k^10 + 410456*k^8 + 383856*k^6 + 47936*k^4 + 256*k^2)*x^10/10! + (2702765*k^12 + 32947964*k^10 + 54480944*k^8 + 17142784*k^6 + 780544*k^4 + 1024*k^2)*x^12/12! + (199360981*k^14 + 3402510924*k^12 + 8760740640*k^10 + 5199585280*k^8 + 686711040*k^6 + 12555264*k^4 + 4096*k^2)*x^14/14! + ...
such that D(x,k)^2 - k^2*S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. D(x,k) begins:
1;
0, 1;
0, 4, 5;
0, 16, 148, 61;
0, 64, 2832, 6744, 1385;
0, 256, 47936, 383856, 410456, 50521;
0, 1024, 780544, 17142784, 54480944, 32947964, 2702765;
0, 4096, 12555264, 686711040, 5199585280, 8760740640, 3402510924, 199360981;
0, 16384, 201199616, 26090711040, 419867864320, 1569971730560, 1632067372896, 441239943664, 19391512145;
0, 65536, 3220652032, 965223559168, 30892394850304, 227204970315520, 502094919789184, 353538702361888, 70347660061552, 2404879675441; ...
RELATED SERIES.
The related series S(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
S(x,k) = x + (2*k^2 + 1)*x^3/3! + (16*k^4 + 28*k^2 + 1)*x^5/5! + (272*k^6 + 1032*k^4 + 270*k^2 + 1)*x^7/7! + (7936*k^8 + 52736*k^6 + 36096*k^4 + 2456*k^2 + 1)*x^9/9! + (353792*k^10 + 3646208*k^8 + 4766048*k^6 + 1035088*k^4 + 22138*k^2 + 1)*x^11/11! + (22368256*k^12 + 330545664*k^10 + 704357760*k^8 + 319830400*k^6 + 27426960*k^4 + 199284*k^2 + 1)*x^13/13! + ...
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
C(x,k) = 1 + x^2/2! + (8*k^2 + 1)*x^4/4! + (136*k^4 + 88*k^2 + 1)*x^6/6! + (3968*k^6 + 6240*k^4 + 816*k^2 + 1)*x^8/8! + (176896*k^8 + 513536*k^6 + 195216*k^4 + 7376*k^2 + 1)*x^10/10! + (11184128*k^10 + 51880064*k^8 + 39572864*k^6 + 5352544*k^4 + 66424*k^2 + 1)*x^12/12! + (951878656*k^12 + 6453433344*k^10 + 8258202240*k^8 + 2458228480*k^6 + 139127640*k^4 + 597864*k^2 + 1)*x^14/14! + ...

Links

Crossrefs

Cf. A322230 (S), A322231 (C), A000364 (diagonal), A001818(row sums).
Cf. A325221 (row reversal).

Programs

  • PARI
    N=10;
{S=x;C=1;D=1; for(i=1,2*N, S = intformal(C*D^2 +O(x^(2*N+1))); C = 1 + intformal(S*D^2); D = 1 + k^2*intformal(S*C*D));}
for(n=0,N, for(j=0,n, print1( (2*n)!*polcoeff(polcoeff(D,2*n,x),2*j,k),", ")) ;print(""))

Formula

E.g.f. D = D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n) * k^(2*j) / (2*n)!, along with related series S = S(x,k) and C = C(x,k), satisfies:
(1a) S = Integral C*D^2 dx.
(1b) C = 1 + Integral S*D^2 dx.
(1c) D = 1 + k^2 * Integral S*C*D dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral D^2 dx ).
(3b) D + k*S = exp( k * Integral C*D dx ).
(4a) S = sinh( Integral D^2 dx ).
(4b) S = sinh( k * Integral C*D dx ) / k.
(4c) C = cosh( Integral D^2 dx ).
(4d) D = cosh( k * Integral C*D dx ).
(5a) d/dx S = C*D^2.
(5b) d/dx C = S*D^2.
(5c) d/dx D = k^2 * S*C*D.
From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral D dx, k),
(6b) C = cn( i * Integral D dx, k),
(6c) D = dn( i * Integral D dx, k).
(7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
(7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
(7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Main diagonal equals A000364, the secant numbers.

A354115 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arcsin(x).

Original entry on oeis.org

1, -2, 1, -4, 29, -244, 1583, -10368, 124553, -2029776, 20127867, -180343296, 3978820221, -75977108544, 914656587063, -15574206480384, 370244721585681, -8082505243732224, 162968423791332339, -3082360882836013056, 82014901819948738629, -2501342802748968883200, 58311771938510122952559
Offset: 1

Views

Author

Ilya Gutkovskiy, May 17 2022

Keywords

Crossrefs

Cf. A001818, A353818, A353913, A354055, A354116, A354117, A354118.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcSin[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

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

A354116 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, -2, -1, 4, -11, 116, -547, 960, -7751, 414384, -3258663, -6813696, -390445563, 9694641984, -964154427, 208258646016, -18431412645519, 207842731632384, -6436900596281679, -37454668211552256, 834261829219880829, 91517388643567641600, -1149793471388581053219
Offset: 1

Views

Author

Ilya Gutkovskiy, May 17 2022

Keywords

Crossrefs

Cf. A001818, A296435, A353819, A353914, A354056, A354115, A354117, A354118.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcSinh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

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

A012248 Expansion of e.g.f. exp(arcsinh(arcsin(x))).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 49, 225, 897, 11025, 96801, 893025, 6803457, 108056025, 1275363153, 18261468225, 207592347393, 4108830350625, 60889593787713, 1187451971330625, 17888210916886017, 428670161650355625, 7679611833095218545, 189043541287806830625, 3530100224793651058305
Offset: 0

Views

Author

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

Keywords

Examples

			1 + x + 1/2!*x^2 + 1/3!*x^3 + 1/4!*x^4 + 9/5!*x^5...

Crossrefs

Bisections are |A012115(n)| and A001818.

Programs

  • Mathematica
    With[{nn=25},CoefficientList[Series[Exp[ArcSinh[ArcSin[x]]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 02 2020 *)
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(exp(asinh(asin(x))))) \\ Michel Marcus, Mar 09 2017

Formula

E.g.f.: Q(0)-1, where Q(k) = 2 + arcsin(x)/(1 - arcsin(x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2013

Extensions

More terms from Michel Marcus, Mar 09 2017

A291482 Expansion of e.g.f. arcsin(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 4, 8, 24, 80, 456, 2368, 20352, 139648, 1577984, 13327360, 185992832, 1860708096, 30882985472, 356724338688, 6860887896064, 89815091306496, 1963843714723840, 28724760194564096, 703639672161697792, 11370790299166343168, 308435832182144040960, 5456591088206554333184, 162354575283061816197120
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 24*x^5/5! + ...

Crossrefs

Cf. A001818, A009545, A012316, A081919 (first differences).

Programs

  • Maple
    a:=series(arcsin(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSin[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Exp[x] x Sqrt[1 - x^2]/(1 + ContinuedFractionK[-2 x^2 Floor[(k + 1)/2] (2 Floor[(k + 1)/2] - 1), 2 k + 1, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Sum[(x^(2 k + 1) Pochhammer[1/2, k])/(k! + 2 k k!), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n,2k+1]Binomial[2k,k] (2k)!/4^k,{k,0,(n-1)/2}],{n,0,12}] (* Emanuele Munarini, Dec 17 2017 *)
  • Maxima
    makelist(sum(binomial(n,2*k+1)*binomial(2*k,k)*(2*k)!/4^k,k,0,floor((n-1)/2)),n,0,12); /* Emanuele Munarini, Dec 17 2017 */
  • PARI
    x='x+O('x^99); concat(0, Vec(serlaplace(asin(x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

Formula

E.g.f.: exp(x)*x*sqrt(1 - x^2)/(1 - 1*2*x^2/(3 - 1*2*x^2/(5 - 3*4*x^2/(7 - 3*4*x^2/(9 - ...))))), a continued fraction.
a(n) ~ (exp(2) - (-1)^n) * n^(n-1) / exp(n+1). - Vaclav Kotesovec, Aug 26 2017
From Emanuele Munarini, Dec 17 2017: (Start)
a(n) = Sum_{k=0..(n-1)/2} binomial(n,2*k+1)*binomial(2*k,k)* (2k)!/4^k.
a(n+4) - 2*a(n+3) - (n^2+4*n+3)*a(n+2) + (n+2)*(2*n+3)*a(n+1) - (n+1)*(n+2)*a(n) = 0. (End)

A296462 Expansion of e.g.f. arcsin(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 12, 238, 9912, 708282, 77392260, 12002011110, 2507167177200, 678724656721650, 231129344455890300, 96694934804540934750, 48752132066414189721000, 29154453671147281799726250, 20403607225475633039372992500, 16520371586328834323725749873750, 15322889489994265975004588078700000
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsin(x)*arctanh(x) = 2*x^2/2! + 12*x^4/4! + 238*x^6/6! + 9912*x^8/8! + 708282*x^10/10! + ...

Crossrefs

Cf. A001818, A009744, A009747, A010050, A012723, A296463.

Programs

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

Formula

E.g.f.: arcsinh(x)*arctan(x) (even powers only, absolute values).
E.g.f.: i*(log(1 - x) - log(1 + x))*log(i*x + sqrt(1 - x^2))/2, where i is the imaginary unit (even powers only).
a(n) ~ Pi * (2*n-1)! / 2. - Vaclav Kotesovec, Dec 13 2017

A296741 Expansion of e.g.f. arcsin(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 4, 64, 2752, 237312, 34390016, 7512117248, 2302977392640, 942529341030400, 496287845973753856, 326775812392982937600, 263039306566659448242176, 254121613033387345942937600, 290175686081926976733941071872, 386599796043915196967089006968832
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsin(x*sec(x)) = x/1! + 4*x^3/3! + 64*x^5/5! + 2752*x^7/7! + 237312*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296742, A296743.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*sec(x)).

A296742 Expansion of e.g.f. arcsinh(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 2, 4, -8, 2448, 130976, -2342848, -239130240, 99052990720, 8918588764672, -2795242017684480, -92786315822417920, 279479081010906828800, -57316070780459900928, -39411396653183724314673152, 5932051008707372732672475136, 10689040617354387626585873252352
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsinh(x*sec(x)) = x/1! + 2*x^3/3! + 4*x^5/5! - 8*x^7/7! + 2448*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296741, A296743.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asinh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x*sec(x)).
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