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 1-7 of 7 results.

A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 10, 1, 0, 280, 120, 1, 15400, 17160, 2080, 1401401, 3203200, 1290640, 190623040, 775975201, 712150400, 36321556720, 239000886400, 413465452401, 9339501072000, 91625659447400, 266045692290560, 3216459513124001, 42923384190336000, 193108117771690680
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Number of partitions of n-set into blocks congruent to 3 mod 4.

Links

Crossrefs

Cf. A002017, A003724, A004767, A005046, A013369, A035462, A291975, A306347, A307979.

Programs

  • Mathematica
    nmax = 29; CoefficientList[Series[Exp[(Sinh[x] - Sin[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{3}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sinh(x)-sin(x))/2))) \\ Seiichi Manyama, Mar 17 2022
  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-3)\4, binomial(n-1, 4*k+2)*a(n-4*k-3))); \\ Seiichi Manyama, Mar 17 2022

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n-1,4*k+2) * a(n-4*k-3). - Seiichi Manyama, Mar 17 2022

A352429 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n,4*k+1) * a(n-4*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 732, 5166, 41664, 378001, 3810512, 42253926, 511139904, 6698457481, 94535404992, 1429477706286, 23056267551744, 395120495014561, 7169579673404672, 137321623511274246, 2768602189953629184, 58609968225266985241, 1299827736206335767552, 30137364376923272989806
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2022

Keywords

Links

Crossrefs

Cf. A000670, A006154, A243665, A291975, A306347, A352428, A352430.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 4 k + 1] a[n - 4 k - 1], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(4 k + 1)/(4 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\4, x^(4*k+1)/(4*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

Formula

E.g.f.: 1 / (1 - Sum_{k>=0} x^(4*k+1) / (4*k+1)!).
E.g.f.: 1 / (1 - (sin(x) + sinh(x)) / 2).

A333882 Expansion of e.g.f. exp(Sum_{k>=0} x^(5*k + 1) / (5*k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 1399, 7801, 45410, 216581, 853218, 2896002, 11708734, 79817500, 615700986, 4012571831, 21538473686, 98707812691, 501634082800, 3983368886226, 37404203343457, 305886831698593, 2069143637726674, 11924094649669375
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

Comments

Number of partitions of n-set into blocks congruent to 1 mod 5.

Crossrefs

Cf. A000110, A003724, A306347, A333881, A333883.

Programs

  • Mathematica
    nmax = 29; CoefficientList[Series[Exp[Sum[x^(5 k + 1)/(5 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 5]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {2/5, 3/5, 4/5, 6/5}, x^5/3125]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} binomial(n-1,5*k) * a(n-5*k-1). - Seiichi Manyama, Sep 22 2023

A013025 Expansion of e.g.f. exp(sinh(x) + sin(x)).

Original entry on oeis.org

1, 2, 4, 8, 16, 34, 88, 296, 1152, 4546, 17696, 72712, 343424, 1843170, 10274688, 56506024, 315332608, 1910439298, 12815815168, 90064672520, 629185325056, 4400756254114, 32422278027264, 258933905154856, 2168521319694336
Offset: 0

Views

Author

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

Keywords

Examples

			1+2*x+4/2!*x^2+8/3!*x^3+16/4!*x^4+34/5!*x^5...

Links

Crossrefs

Cf. A013369, A306347.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Sinh[x]+Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 17 2011 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sin(x)+sinh(x)))) \\ Seiichi Manyama, Mar 17 2022
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - Seiichi Manyama, Mar 17 2022

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3*k + 1) / (3*k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148579, 758916, 3580189, 18981485, 117883917, 720627553, 4193077474, 26795418840, 191751387094, 1352954503595, 9301704998742, 69285817230370, 559142785301527, 4453089770243547, 35182348161102172
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

Comments

Number of partitions of n-set into blocks congruent to 1 mod 3.

Crossrefs

Cf. A000110, A003724, A193437, A306347, A333882, A333883.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 3]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]
nmax = 30; CoefficientList[Series[Exp[Exp[x]/3 - 2*Sin[Pi/6 - Sqrt[3]*x/2] / (3*Exp[x/2])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

Formula

E.g.f.: exp(exp(x)/3 - 2*sin(Pi/6 - sqrt(3)*x/2) / (3*exp(x/2))). - Vaclav Kotesovec, Apr 15 2020
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(n-3*k-1). - Seiichi Manyama, Sep 22 2023

A307979 Expansion of e.g.f. exp((cosh(x) - cos(x))/2) (even powers only).

Original entry on oeis.org

1, 1, 3, 16, 133, 1576, 24783, 495496, 12245353, 364768576, 12838252443, 526095538816, 24781014246253, 1326767681420416, 80013978835916583, 5392682199766283776, 403287063337529642833, 33261775377836063850496, 3009257393136250807614003, 297176659119237977183973376
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Number of partitions of a 2n-set into blocks congruent to 2 mod 4.

Crossrefs

Cf. A003724, A005046, A013469, A016825, A035457, A291975, A306347, A307978.

Programs

  • Mathematica
    nmax = 19; Table[(CoefficientList[Series[Exp[(Cosh[x] - Cos[x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[n_] := a[n] = Sum[Boole[MemberQ[{2}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[2 n], {n, 0, 19}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp((cosh(x) - cos(x))/2).

A333883 Expansion of e.g.f. exp(Sum_{k>=0} x^(6*k + 1) / (6*k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 5163, 32281, 217921, 1188709, 5291353, 20031170, 66744741, 267996541, 2030569465, 18368560519, 138812739409, 853152218102, 4409607501927, 19826125988257, 99717123889777, 871344991322017, 9658479225877057
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

Comments

Number of partitions of n-set into blocks congruent to 1 mod 6.

Crossrefs

Cf. A000110, A003724, A306347, A333881, A333882.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[Sum[x^(6 k + 1)/(6 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 6]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {1/3, 1/2, 2/3, 5/6, 7/6}, x^6/46656]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} binomial(n-1,6*k) * a(n-6*k-1). - Seiichi Manyama, Sep 22 2023
Showing 1-7 of 7 results.

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