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.

Showing 1-9 of 9 results.

A296464 Expansion of e.g.f. arcsin(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 2, 28, 1024, 71632, 8192736, 1392793920, 330041217024, 104069101383936, 42159457593506304, 21346870862961183744, 13213529766600134344704, 9818417126704155249954816, 8625630408510010165396070400, 8844234850947343105068735283200, 10467364426053362392901751845683200
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsin(arcsin(x)) = x/1! + 2*x^3/3! + 28*x^5/5! + 1024*x^7/7! + 71632*x^9/9! + ...

Crossrefs

Cf. A001818, A003712, A012063, A296466.

Programs

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

Formula

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

A003715 Expansion of e.g.f. sin(sin(sin(x))) (odd powers only).

Original entry on oeis.org

1, -3, 33, -731, 25857, -1311379, 89060065, -7778778091, 849264442881, -113234181108643, 18073465545032353, -3395124358886313595, 740061366713642835201, -185005977382236600650035
Offset: 0

Views

Author

R. H. Hardin, Simon Plouffe

Keywords

Comments

{abs(a(n))} has e.g.f. sinh(sinh(sinh(x))) (odd powers only). - Jianing Song, Oct 25 2023

References

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

Links

Crossrefs

Cf. A003712.

Programs

  • Mathematica
    With[{nn = 50}, Take[CoefficientList[Series[Sin[Sin[Sin[x]]], {x, 0, nn}], x] Range[0, nn - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)
  • Maxima
    a(k):=sum((sum((2^(-2*j-2*m)*(sum((2*i-2*j-1)^(2*m+1)*(-1)^(i)*binomial(2*j+1,i),i,0,(2*j+1)/2))*sum((2*i-2*m-1)^(2*k+1)*binomial(2*m+1,i)*(-1)^(k-i),i,0,(2*m+1)/2))/(2*m+1)!,m,j,k))/(2*j+1)!,j,0,k); /* Vladimir Kruchinin, Jun 10 2011 */

Formula

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

A358834 Number of odd-length twice-partitions of n into odd-length partitions.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 11, 24, 35, 74, 109, 213, 336, 624, 986, 1812, 2832, 5002, 7996, 13783, 21936, 37528, 59313, 99598, 158356, 262547, 415590, 684372, 1079576, 1759984, 2779452, 4491596, 7069572, 11370357, 17841534, 28509802, 44668402, 70975399, 110907748
Offset: 0

Views

Author

Gus Wiseman, Dec 04 2022

Keywords

Comments

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (111)      (211)      (221)            (222)
            (1)(1)(1)  (2)(1)(1)  (311)            (321)
                                  (11111)          (411)
                                  (2)(2)(1)        (21111)
                                  (3)(1)(1)        (2)(2)(2)
                                  (111)(1)(1)      (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (111)(2)(1)
                                                   (211)(1)(1)
                                                   (2)(1)(1)(1)(1)

Links

Crossrefs

The version for set partitions is A003712.
If the parts are also odd we get A279374.
The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
This is the odd-length case of A358334.
This is the odd-lengths case of A358824.
For odd sums instead of lengths we have A358826.
The case of odd sums also is the bisection of A358827.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A270995, A271619, A279785, A358823, A358837.

Programs

  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]],{n,0,10}]
  • PARI
    P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

Formula

G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A359553 Numerator of the coefficient of x^(2n+1) in the Taylor series expansion of sin(sin(x)).

Original entry on oeis.org

1, -1, 1, -8, 13, -47, 15481, -15788, 451939, -23252857, 186846623, -831520891, 1108990801, -143356511198507, 920716137922619, -13390469094133441, 929480267163260699, -118186323448146684881, 69875813865886026036091, -155759565768613453511731
Offset: 0

Views

Author

Kevin Ryde, Jan 09 2023

Keywords

Comments

Denominators are A359554.
Sine is an odd function so the Taylor series has 0 coefficients at even terms x^(2n).
A003712(n) is the numerator for use with denominator (2n+1)! so that here a(n)/A359554(n) = A003712(n)/(2n+1)! reduced to least terms.
abs(a(n)) is the corresponding numerator in the expansion of sinh(sinh(x)).

Examples

			Fractions begin: 1, -1/3, 1/10, -8/315, 13/2520, -47/49896, ...
Series begins: sin(sin(x)) = x - (1/3)*x^3 + (1/10)*x^5 - (8/315)*x^7 + ...

Links

Crossrefs

Cf. A359554 (denominators), A003712 (e.g.f. sin(sin(x))).

Programs

  • PARI
    a_vector(len) = apply(numerator, Vec(substpol(sin(sin(Ser('x,,2*len)))/'x, 'x^2,'x)));

Formula

a(n) = numerator of A003712(n)/(2n+1)!.
Sum_{n>=0} a(n)/A359554(n) * x^(2*n+1). = sin(sin(x)).
Sum_{n>=0} abs(a(n))/A359554(n) * x^(2*n+1). = sinh(sinh(x)).

A359554 Denominator of the coefficient of x^(2n+1) in the Taylor series expansion of sin(sin(x)).

Original entry on oeis.org

1, 3, 10, 315, 2520, 49896, 97297200, 638512875, 126309456000, 47517617347200, 2934911659680000, 105192125402364000, 1178623253808000000, 1329207696584271504000000, 77094046401887747232000000, 10455880043256025718340000000, 6973521679375808310673920000000
Offset: 0

Views

Author

Kevin Ryde, Jan 11 2023

Keywords

Comments

Numerators are A359553, see there for details.

Links

Crossrefs

Cf. A359553 (numerators), A003712 (e.g.f. sin(sin(x))).

Programs

  • PARI
    a_vector(len) = apply(denominator, Vec(substpol(sin(sin(Ser('x,,2*len)))/'x, 'x^2,'x)));

Formula

a(n) = denominator of A003712(n)/(2n+1)!.

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

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

A366834 Square array read by descending antidiagonals: (-1)^n*T(n,k)/n! is the coefficient of x^(2*n+1) in the Taylor expansion of the k-th iteration of sin(x).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 12, 1, 0, 1, 4, 33, 128, 1, 0, 1, 5, 64, 731, 1872, 1, 0, 1, 6, 105, 2160, 25857, 37600, 1, 0, 1, 7, 156, 4765, 121600, 1311379, 990784, 1, 0, 1, 8, 217, 8896, 368145, 10138880, 89060065, 32333824, 1, 0, 1, 9, 288, 14903, 873936, 42807605, 1162426880, 7778778091, 1272660224, 1, 0
Offset: 0

Views

Author

Jianing Song, Oct 25 2023

Keywords

Comments

T(n,k)/n! is the coefficient of x^(2*n+1) in the Taylor expansion of the k-th iteration of sinh(x). This is most easily seen from the relation -i*sin(...sin(sin(sin(i*x)))...) = -i*sin(...sin(sin(i*sinh(x)))...) = -i*sin(...sin(i*sinh(sinh(x)))...) = ... = sinh(...sinh(sinh(sinh(x)))...).

Examples

			G.f.s of the first few rows:
n = 0: 1/(1-x);
n = 1: x/(1-x)^2;
n = 2: x/(1-x)^2 + 10*x^2/(1-x)^3;
n = 3: x/(1-x)^2 + 126*x^2/(1-x)^3 + 350*x^3/(1-x)^4:
n = 4: x/(1-x)^2 + 1870*x^2/(1-x)^3 + 20244*x^3/(1-x)^4 + 29400*x^4/(1-x)^5;
n = 5: x/(1-x)^2 + 37598*x^2/(1-x)^3 + 1198582*x^3/(1-x)^4 + 5118960*x^4/(1-x)^5 + 4851000*x^5/(1-x)^6.
The explicit formulas for the first few rows:
T(0,k) = binomial(k,0) = 1 for k = 0, 0 for k > 0;
T(1,k) = binomial(k,1) = k;
T(2,k) = binomial(k,1) + 10*binomial(k,2) = 5*k^2 - 4*k;
T(3,k) = binomial(k,1) + 126*binomial(k,2) + 350*binomial(k,3) = (175/3)*k^3 - 112*k^2 + (164/3)*k;
T(4,k) = binomial(k,1) + 1870*binomial(k,2) + 20244*binomial(k,3) + 29400*binomial(k,4) = 1225*k^4 - 3976*k^3 + 4288*k^2 - 1536*k;
T(5,k) = binomial(k,1) + 37598*binomial(k,2) + 1198582*binomial(k,3) + 5118960*binomial(k,4) + 4851000*binomial(k,5) = 40425*k^5 - 190960*k^4 + (1004696/3)*k^3 - 255552*k^2 + (213568/3)*k.
Table of terms:
Row 0: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Row 1: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Row 2: 0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460
Row 3: 0, 1, 128, 731, 2160, 4765, 8896, 14903, 23136, 33945, 47680
Row 4: 0, 1, 1872, 25857, 121600, 368145, 873936, 1776817, 3244032, 5472225, 8687440
Row 5: 0, 1, 37600, 1311379, 10138880, 42807605, 130426016, 323774535, 698156544, 1358249385, 2442955360
Row 6: 0, 1, 990784, 89060065, 1162426880, 6937805945, 27344158016, 83303826249, 212901058560, 478937915985, 977877567040
Row 7: 0, 1, 32333824, 7778778091, 174394695680, 1487589904205, 7634965431296, 28668866786679, 87104014381056, 227079171721785, 527214112015360
Row 8: 0, 1, 1272660224, 849264442881, 33044097597440, 406373544070945, 2731282112246016, 12688038285458529, 45949019179646976, 139088689115885505, 367745105831952640
Row 9: 0, 1, 59527313920, 113234181108643, 7701145601933312, 137461463840219237, 1215573962763120128, 7008667055272520967, 30322784763588771840, 106757902382656031049, 321859857651846029824
Row 10: 0, 1, 3252626013184, 18073465545032353, 2162675327569362944, 56311245536706922889, 657730167421332884480, 4719958813316934631353, 24445625744089126797312, 100254353682662263787313, 345053755346367061654528
Demonstration of terms:
sin(sin(x)) = x - 2*x^3/3! + 12*x^5/5! - 128*x^7/7! + 1872*x^9/9! - 37600*x^11/11! + ...;
sin(sin(sin(x))) = x - 3*x^3/3! + 33*x^5/5! - 731*x^7/7! + 25857*x^9/9! - 1311379*x^11/11! + ...;
sin(sin(sin(sin(x)))) = x - 4*x^3/3! + 64*x^5/5! - 2160*x^7/7! + 121600*x^9/9! - 10138880*x^11/11! + ....

Links

Crossrefs

Cf. A051624 (row n=2), A366827 (row n=3), A003712 (column k=2 signed), A003715 (column k=3 signed).
Cf. A000447, A285018, A285019.

Programs

  • PARI
    A160562(n,k) = (-1)^k / (2*k+1)! * sum(j=0, k, (-1)^j * binomial(2*k+1, k-j) * (2*j+1)^(2*n+1)) / 2^(2*k)
coeff_of_n_gfs(n) = my(M=matid(1)); for(k=1, n, M = matconcat([concat(M, matrix(k, 1)); concat(0, matrix(1, k, i, j, A160562(k, j-1))*M)])); M \\ The lower triangle matrix (C(i,j))_{0<=j<=i<=n}
T_mat(n,k) = coeff_of_n_gfs(n)*matrix(n+1, k+1, i, j, binomial(j-1,i-1)) \\ gives T(i,j) for i=0..n and j=0..k

Formula

T(0,0) = 1, T(n,0) = 0 for n >= 1; T(n,k) = Sum_{i=0..n} A160562(n,i)*T(i,k-1) for k >= 1, where A160562(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1). Note that this is not very efficient to calculate the terms.
A more efficient way would be to calculate the g.f. for each row: the g.f. of the n-th row is C(n,0)/(1-x) + C(n,1)*x/(1-x)^2 + ... + C(n,n)*x^n/(1-x)^(n+1), where C(0,0) = 1, C(n,0) = 0 for n >= 1; C(n,k) = A160562(n,k-1)*C(k-1,k-1) + ... + A160562(n,n-1)*C(n-1,k-1) for n >= k >= 1, so we have T(n,k) = C(n,0)*binomial(k,0) + C(n,1)*binomial(k,1) + ... + C(n,n)*binomial(k,n). See my pdf in the link section for the proof.
From the formula above we see that the n-th row is a degree-n polynomial of k with leading coefficient C(n,n)/n!. We have C(n,n) = A160562(n,n-1)*C(n-1,n-1) = A000447(n)*C(n-1,n-1) for n >= 1, so it can be shown that C(n,n)/n! = n! * A285018(n)/A285019(n).

A212261 Array A(i,j) read by antidiagonals: A(i,j) is the (2i-1)-th derivative of sin(sin(sin(...sin(x)))) nested j times evaluated at 0.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -3, 12, -1, 1, -4, 33, -128, 1, 1, -5, 64, -731, 1872, -1, 1, -6, 105, -2160, 25857, -37600, 1, 1, -7, 156, -4765, 121600, -1311379, 990784, -1, 1, -8, 217, -8896, 368145, -10138880, 89060065, -32333824, 1
Offset: 1

Views

Author

John M. Campbell, May 12 2012

Keywords

Comments

The determinant of the n X n such matrix has a closed form given in the formula section (and the Mathematica code below).
Rows appear to be given by polynomials (see formula section).

Examples

			Evaluate the fifth derivative of sin(sin(sin(x))) at 0, which is 33. So the (3,3) entry of the array is 33. The array begins as:
|  1      1        1         1         1          1 |
| -1     -2       -3        -4        -5         -6 |
|  1     12       33        64       105        156 |
| -1   -128     -731     -2160     -4765      -8896 |
|  1   1872    25857    121600    368145     873936 |
| -1 -37600 -1311379 -10138880 -42807605 -130426016 |

Crossrefs

Cf. A003712, A003715.

Programs

  • Maple
    A:= (i, j)-> (D@@(2*i-1))(sin@@j)(0):
seq(seq(A(i, 1+d-i), i=1..d), d=1..9); # Alois P. Heinz, May 14 2012
  • Mathematica
    A[a_, b_] :=
  A[a, b] =
   Array[D[Nest[Sin, x, #2], {x, 2*#1 - 1}] /. x -> 0 &, {a, b}];
Print[A[7, 7] // MatrixForm];
Table2 = {};
k = 1;
While[k < 8, Table1 = {};
  i = 1;
  j = k;
  While[0 < j,
   AppendTo[Table1, First[Take[First[Take[A[7, 7], {i, i}]], {j, j}]]];
   j = j - 1;
   i = i + 1];
  AppendTo[Table2, Table1];
  k++];
Print[Flatten[Table2]]
Print[Table[Det[A[n, n]], {n, 1, 7}]];
Print[Table[(
  I^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2)
    Glaisher^3 (-(1/\[Pi]))^
   n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]]

Formula

A(i,j) = ((d/dx)^(2i-1) sin^j(x))_{x=0}.
Let A_n denote the n X n such matrix. Then:
det(A_n)=(i^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2) G^3 (-(1/pi))^n B(1/2 + n) B(1 + n) B(3/2 + n))/e^(1/4), where B is the Barnes G-function and G is the Glaisher-Kinkelin constant (and i is the imaginary unit). (This can be shown by evaluating recurrence relations for det(A_n)). See Mathematica code below.
First row: 1.
Second row: -x.
Third row: x (5 x - 4).
Fourth row: -(1/3) x (164 + 7 x (-48 + 25 x)).
Fifth row: (8 - 7 x)^2 x (-24 + 25 x).
Sixth row: -(1/3) x (213568 - 766656 x + 1004696 x^2 - 572880 x^3 + 121275 x^4).
Seventh row: 1/3 x (-14371328 + 65012064 x - 111160192 x^2 + 91291200 x^3 - 36552516 x^4 + 5780775 x^5).
Second column: A003712.
Third column: A003715.

A302452 a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. sinh(x).

Original entry on oeis.org

1, 2, 33, 2160, 368145, 130426016, 83303826249, 87104014381056, 139088689115885505, 321859857651846029824, 1036109938469605247521009, 4490275483028481600517832704, 25503692273369769781221175069521, 185636732310716855091866841134243840, 1699077450890747555020338066545506541145
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 08 2018

Keywords

Comments

a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. sin(x) (absolute values).

Examples

			The initial coefficients of successive iterations of e.g.f. A(x) = sinh(x) (odd powers only) are as follows:
n = 1: (1), 1,    1,     1,       1,  ... e.g.f. A(x)
n = 2:  1, (2),  12,   128,    1872,  ... e.g.f. A(A(x))
n = 3:  1,  3,  (33),  731,   25857,  ... e.g.f. A(A(A(x)))
n = 4:  1,  4,   64, (2160)  121600,  ... e.g.f. A(A(A(A(x))))
n = 5:  1,  5,  105,  4765, (368145), ... e.g.f. A(A(A(A(A(x)))))
...
More explicitly, the successive iterations of e.g.f. A(x) = sinh(x) begin:
sinh(x) = x/1! + x^3/3! + x^5/5! + x^7/7! + x^9/9! + ...
sinh(sinh(x)) = x/1! + 2*x^3/3! + 12*x^5/5! + 128*x^7/7! + 1872*x^9/9! + ...
sinh(sinh(sinh(x))) = x/1! + 3*x^3/3! + 33*x^5/5! + 731*x^7/7! + 25857*x^9/9! + ...
sinh(sinh(sinh(sinh(x)))) = x/1! + 4*x^3/3! + 64*x^5/5! + 2160*x^7/7! + 121600*x^9/9! + ...
sinh(sinh(sinh(sinh(sinh(x))))) = x/1! + 5*x^3/3! + 105*x^5/5! + 4765*x^7/7! + 368145*x^9/9! + ...

Links

Crossrefs

Cf. A003712, A003715.

Programs

  • Mathematica
    Table[(2 n - 1)! SeriesCoefficient[Nest[Function[x, Sinh[x]], x, n], {x, 0, 2 n - 1}], {n, 15}]
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