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 17 results. Next

A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

Original entry on oeis.org

1, 1, 2, 9, 48, 305, 2400, 22057, 230272, 2708001, 35412480, 509177801, 7986468864, 135718942801, 2483729876992, 48699677975145, 1018542257111040, 22634000289407297, 532557637644976128, 13226748101381102473, 345792863300174479360, 9492229607399841038961
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2012

Keywords

Comments

Radius of convergence of e.g.f. is |x| < r where r = 0.7650099545507... satisfies cosh(r) = 1/r. See A069814.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! +...

Links

Crossrefs

Cf. A069814, A185951.
Cf. A000111, A006154, A352252.
Cf. A381206, A381207.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1-x*cosh(x +x*O(x^n))),n)}
  • 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!*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(2*n-1) == 1 (mod 4), a(2*n+2) == 0 (mod 4), for n>=1.
a(n) ~ n!/(1+r*sqrt(1-r^2))*(1/r)^n, where r = A069814 = 0.7650099545507321... is the root of the equation r*cosh(r)=1. - Vaclav Kotesovec, Feb 13 2013
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} k! * A185951(n,k). - Seiichi Manyama, Feb 17 2025

A009189 Expansion of e.g.f.: exp(cos(x)*x).

Original entry on oeis.org

1, 1, 1, -2, -11, -24, 61, 624, 1737, -7424, -88679, -242560, 2086525, 23499776, 45950997, -1002251264, -9763133167, -2151563264, 705668046769, 5583112077312, -17356978593659, -666018502836224, -3823112141007763, 39230927775531008, 788728947108214489
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A003727, A352252.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Cos[x]*x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 15 2018 *)
  • Maxima
    a(n):=(sum(binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(binomial(k,i)*(k-2*i)^(n-k),i,0,floor((k-1)/2)),k,1,n-1))+1; /* Vladimir Kruchinin, Apr 21 2011 */
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*cos(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

a(n) = (sum(k=1..n-1, binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(i=0..floor((k-1)/2)), binomial(k,i)*(k-2*i)^(n-k)))+1. - Vladimir Kruchinin, Apr 21 2011
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Mar 15 2018

A385310 Expansion of e.g.f. 1/(1 - 2 * x * cos(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 12, 69, 500, 4455, 46928, 571977, 7914384, 122585355, 2100940864, 39470867469, 806555184448, 17808628411119, 422498774818560, 10717948285126545, 289501146405400832, 8295124400250875667, 251300745071590317056, 8025654235707259740885, 269482309052945201181696
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385311.
Cf. A380015, A385304, A385306, A385308.
Cf. A001147, A185951, A352646, A385283.

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)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A385311 Expansion of e.g.f. 1/(1 - 3 * x * cos(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 25, 232, 2805, 41920, 744933, 15340416, 359136073, 9419223040, 273558859409, 8714789788672, 302151400126589, 11326084055150592, 456421403198919325, 19677025400034590720, 903660903945306053137, 44042354270955276599296, 2270411632567521580120713
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385310.
Cf. A380017, A385305, A385307, A385309.
Cf. A007559, A185951, A352647, A385284.

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)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A381209 Expansion of e.g.f. 1/(1 - x*cos(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 216, 735, 0, -39081, -575232, -6047973, -48314880, -189159333, 3046957056, 99745485879, 1789140627456, 23433663134655, 185580069027840, -1250544374605389, -94781673979379712, -2543434372808424957, -47763303489939701760, -586864592847636893937
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, A381207, A381210, A381211.
Cf. A352252, A381208.
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)!*I^(n-k)*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381283 Expansion of e.g.f. 1/(1 - x * cos(3*x)).

Original entry on oeis.org

1, 1, 2, -21, -192, -1095, 7200, 243747, 3088512, 1360881, -874437120, -21701765349, -186175604736, 5870711879721, 292185085151232, 5507319584787795, -38951106749890560, -6402114772676575263, -212680600451474522112, -1602903494245708491957, 197042528380347210792960
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. A352252, A381282.
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!*(3*I)^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (3*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A349104 Expansion of e.g.f. 1/(1 - (sin(x) + x*cos(x))/2 ).

Original entry on oeis.org

1, 1, 2, 4, 8, 3, -124, -1306, -10144, -67723, -363392, -831672, 16709824, 386800759, 5631873664, 66256305994, 619010054144, 3201069236265, -40479063835648, -1775812586063860, -39853546353553408, -694055641682352469, -9591063643658387456, -84103588142498507346
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2022

Keywords

Links

Crossrefs

Cf. A349103, A349105, A352252.

Programs

  • Mathematica
    With[{m = 23}, Range[0, m]! * CoefficientList[Series[1/(1 - (Sin[x] + x*Cos[x])/2), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-(sin(x)+x*cos(x))/2)))
  • PARI
    a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (-1)^k*(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)} (-1)^k * (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).

A352646 Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

Original entry on oeis.org

1, 2, 8, 42, 288, 2410, 24000, 277186, 3648512, 53936082, 885150720, 15970846298, 314273439744, 6698574264122, 153746319720448, 3780677636321010, 99163499845386240, 2763481838977368994, 81542013760903053312, 2539717324111483027594
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A352252, A352647.
Cf. A352642, A352648.
Cf. A185951.

Programs

  • Mathematica
    With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cos[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*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(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)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 - r * sqrt(4*r^2 - 1)) * r^n), where r = A196603 = 0.6100312844641759753709630735134103246737209791121692378637516075328... is the root of the equation 2*r*cos(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A352647 Expansion of e.g.f. 1/(1 - 3 * x * cos(x)).

Original entry on oeis.org

1, 3, 18, 153, 1728, 24315, 410400, 8079729, 181786752, 4601232243, 129402385920, 4003157532297, 135098815002624, 4939266681129963, 194472450526169088, 8203835046344538465, 369151362125290045440, 17649035213360472293091, 893431062200523039178752
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A352252, A352646.
Cf. A352643, A352649.
Cf. A185951.

Programs

  • Mathematica
    With[{m = 18}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025
Showing 1-10 of 17 results. Next

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