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 1-10 of 20 results. Next

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

Original entry on oeis.org

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Links

Crossrefs

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022
  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025

A069814 Decimal expansion of root of the equation x*cosh(x)=1.

Original entry on oeis.org

7, 6, 5, 0, 0, 9, 9, 5, 4, 5, 5, 0, 7, 3, 2, 1, 2, 2, 6, 5, 5, 3, 2, 1, 7, 4, 2, 4, 8, 2, 8, 1, 5, 2, 1, 9, 2, 0, 0, 3, 5, 2, 1, 3, 7, 4, 7, 5, 0, 4, 3, 3, 2, 3, 1, 6, 2, 4, 7, 0, 7, 1, 0, 7, 4, 0, 0, 1, 9, 2, 3, 1, 9, 9, 4, 4, 8, 1, 4, 1, 2, 7, 8, 0, 8, 2
Offset: 0

Views

Author

Benoit Cloitre, Apr 30 2002

Keywords

Examples

			0.76500995455073212265532174248281521920035213747504332316247071...

Links

Crossrefs

Cf. A205571, A385281, A385282.

Programs

  • Maple
    Digits:= 120:
fsolve(x*cosh(x)=1,x=0..1); # Robert Israel, Jan 07 2015
  • Mathematica
    RealDigits[x/.FindRoot[x*Cosh[x]==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 06 2015 *)
  • PARI
    solve(x=0, 1, x*cosh(x)-1) \\ Michel Marcus, Jan 06 2015

Extensions

Corrected by Harvey P. Dale, Jan 06 2015

A381207 Expansion of e.g.f. 1/(1 - x*cosh(x))^3.

Original entry on oeis.org

1, 3, 12, 69, 504, 4335, 43200, 490161, 6220032, 87242427, 1340305920, 22375475133, 403237638144, 7801208775399, 161245892161536, 3545854432602345, 82653484859228160, 2035605515838402291, 52814589875313573888, 1439814136866851346357, 41145786213980645621760
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A377530, A381209, A381210, A381211.
Cf. A205571, A381206.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+2)!*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A185951(n,k).

A381280 Expansion of e.g.f. 1/(1 - x * cosh(2*x)).

Original entry on oeis.org

1, 1, 2, 18, 120, 920, 10320, 126448, 1714048, 27073152, 472354560, 8989147904, 187690331136, 4245706716160, 103239264593920, 2691918892861440, 74885151106498560, 2212607133043884032, 69227613551324233728, 2286465386258267176960, 79487593489348266557440
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A381281.
Cf. A185951, A201939.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*2^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^(n-k) * A185951(n,k).
a(n) ~ sqrt(Pi) * 2^(n + 5/2) * n^(n + 1/2) / ((1 + sinh(r))^2 * exp(n) * r^(n+2)), where r = A201939. - Vaclav Kotesovec, Apr 19 2025

A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 27, 249, 2825, 41355, 708883, 13888497, 309267729, 7698772755, 211585744139, 6367841422569, 208299923870233, 7357493992966299, 279095125351544835, 11316313498670411745, 488403056864943302177, 22355228989851909617187, 1081663315375339026249211
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385282.
Cf. A001586, A235134, A380155, A385283.
Cf. A001147, A185951.
Cf. A069814.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + r*sqrt(1 - r^2)) * exp(n) * r^n), where r = A069814. - Vaclav Kotesovec, Jun 24 2025

A385308 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385309.
Cf. A380015, A385304, A385306, A385310.
Cf. A001147, A185951, A352648, A385281.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + r*sqrt(1 - 4*r^2)) * exp(n) * r^n), where r = 0.452787214835453627588998503316635625709288535855... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Jun 28 2025

A385309 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 31, 328, 4485, 75520, 1509347, 34916224, 917703145, 27011107840, 880133628231, 31451749424128, 1223047891889837, 51414400611438592, 2323391075748100555, 112315439676217262080, 5783449255108473820497, 316034972288791445241856, 18265740423344520141491951
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385308.
Cf. A380017, A385305, A385307, A385311.
Cf. A007559, A185951, A352649, A385282.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * A185951(n,k), where A185951(n,0) = 0^n.

A349105 Expansion of e.g.f. 1/(1 - (sinh(x) + x*cosh(x))/2 ).

Original entry on oeis.org

1, 1, 2, 8, 40, 243, 1796, 15502, 152608, 1690613, 20814208, 281859540, 4163795648, 66636761575, 1148477490304, 21207704998010, 417728195909632, 8742243282090153, 193720478508563456, 4531158728871170080, 111562803180301643776, 2884156736234559267611
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2022

Keywords

Links

Crossrefs

Cf. A006154, A205571, A349104, A351937.

Programs

  • Mathematica
    With[{m = 21}, Range[0, m]! * CoefficientList[Series[1/(1 - (Sinh[x] + x*Cosh[x])/2), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-(sinh(x)+x*cosh(x))/2)))
  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (k+1) * binomial(n,2*k+1) * a(n-2*k-1).

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 28, 966, 62280, 6452650, 980531916, 205438870014, 56760128400016, 19994672935658322, 8746764024725937300, 4651991306703670964518, 2956156902003429777549144, 2212026607642404922284728826, 1925137044528752884360406444380, 1928103808741894922401976601295950
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A009233, A094088, A205571, A210672, A351762, A352250, A352252, A352253.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[1/(1 - x Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

A352648 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x)).

Original entry on oeis.org

1, 2, 8, 54, 480, 5290, 70080, 1083614, 19145728, 380552274, 8404669440, 204182993542, 5411361939456, 155365918497530, 4803852288901120, 159142710151610670, 5623576097060290560, 211138456468635968674, 8393550198348236193792, 352212802264773650385110
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A205571, A352649.
Cf. A185951.

Programs

  • Mathematica
    With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*cosh(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 + r * sqrt(1 - 4*r^2)) * r^n), where r = 0.452787214835453627588998503316635625709288535855800416726... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025
Showing 1-10 of 20 results. Next

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