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.

A295254 Expansion of e.g.f. csch(x)*(1 - sqrt(1 - 4*sinh(x)))/2.

Original entry on oeis.org

1, 1, 4, 31, 352, 5341, 101824, 2341291, 63092992, 1950837241, 68093599744, 2648776394551, 113633946898432, 5330308817264341, 271416230974603264, 14910196369733535811, 879003840976919068672, 55354496206857969062641, 3708594029795800700944384, 263391744037123969891925071
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A136630, A295237, A295255, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(csch(x)*(1-sqrt(1-4*sinh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Csch[x] (1 - Sqrt[1 - 4 Sinh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sinh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2) * 17^(1/4) * n^(n-1) / (exp(n) * (log((1+ sqrt(17))/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A136630(n,k). - Seiichi Manyama, Feb 23 2025

A295255 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cos(x))).

Original entry on oeis.org

1, 1, 4, 27, 288, 4145, 75360, 1655003, 42628096, 1260274689, 42070233600, 1565308844539, 64237925148672, 2882670856605553, 140430196702035968, 7380867094885024635, 416320345406371921920, 25084955259883686000257, 1608058868442709001895936, 109278344982307590211482971
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A295237, A295254, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*cos(x))),x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Cos[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Cos[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - x*cos(x)/(1 - x*cos(x)/(1 - x*cos(x)/(1 - x*cos(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2 - 2*r*sqrt(16*r^2 - 1)) * n^(n-1) / (exp(n) * r^n), where r = A196605 = 0.2585985822541894903... is the root of the equation r*cos(r) = 1/4. - Vaclav Kotesovec, Nov 18 2017

A295257 Expansion of e.g.f. cot(x)*(1 - sqrt(1 - 4*tan(x)))/2.

Original entry on oeis.org

1, 1, 4, 32, 368, 5656, 109024, 2533712, 68995328, 2155513216, 76014982144, 2987332904192, 129473128921088, 6135478762187776, 315609465774936064, 17515027337549545472, 1043104219010147483648, 66358462250378681614336, 4491141928841064201846784, 322219449242531127348887552
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Links

Crossrefs

Cf. A000108, A195727, A295237, A295254, A295255, A295256, A295258.

Programs

  • Maple
    S:= series(cot(x)*(1 - sqrt(1 - 4*tan(x)))/2, x, 32):
seq(n!*coeff(S,x,n),n=0..30); # Robert Israel, Nov 18 2017
  • Mathematica
    nmax = 19; CoefficientList[Series[Cot[x] (1 - Sqrt[1 - 4 Tan[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Tan[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(17/2) * n^(n-1) / (exp(n) * (arctan(1/4))^(n-1/2)). - Vaclav Kotesovec, Nov 18 2017

A295237 Expansion of e.g.f. csc(x)*(1 - sqrt(1 - 4*sin(x)))/2.

Original entry on oeis.org

1, 1, 4, 29, 320, 4741, 88384, 1988489, 52448000, 1587545161, 54252120064, 2066298252149, 86799115489280, 3986897970744781, 198795278022098944, 10694247962623751009, 617392620634705756160, 38074395493710549747601, 2498063366053169206657024, 173745719989547715852773069
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A195621, A295254, A295255, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(csc(x)*(1-sqrt(1-4*sin(x)))/2,x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Csc[x] (1 - Sqrt[1 - 4 Sin[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sin[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2) * 15^(1/4) * n^(n-1) / (exp(n) * (arcsin(1/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017

A295258 Expansion of e.g.f. coth(x)*(1 - sqrt(1 - 4*tanh(x)))/2.

Original entry on oeis.org

1, 1, 4, 28, 304, 4456, 82144, 1827568, 47674624, 1427337856, 48248157184, 1817752215808, 75534405842944, 3432099993158656, 169290181445914624, 9009094978010165248, 514518446264601739264, 31389459744670699257856, 2037360033664565682110464, 140182487701223036287909888
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A295237, A295254, A295255, A295256, A295257.

Programs

  • Maple
    a:=series(coth(x)*(1-sqrt(1-4*tanh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Coth[x] (1 - Sqrt[1 - 4 Tanh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Tanh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - tanh(x)/(1 - tanh(x)/(1 - tanh(x)/(1 - tanh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(15) * 2^(n-1) * n^(n-1) / (exp(n) * (log(5/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017

A381445 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 75, 1392, 34925, 1108080, 42562807, 1920796416, 99628495353, 5840628226560, 381927689957891, 27562916396961792, 2176123474607538469, 186580455503952427008, 17264834430223073672175, 1714909152672462179205120, 182002038900785304200753777, 20553746198157175799599202304
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

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

Crossrefs

Cf. A205571, A295256, A381446.
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!*binomial(3*k+1, k)/(3*k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(3*k+1,k)/(3*k+1) * A185951(n,k).

A381446 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x) * A(x)^3 ).

Original entry on oeis.org

1, 1, 8, 135, 3456, 120245, 5303040, 283559227, 17830210048, 1289406976713, 105435719470080, 9619902621234191, 968905466782150656, 106779534666615500989, 12781543241568143171584, 1651368425166943566943875, 229049483642619517308764160, 33947359023461155854768564497
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

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

Crossrefs

Cf. A205571, A295256, A381445.
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!*binomial(4*k+1, k)/(4*k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(4*k+1,k)/(4*k+1) * A185951(n,k).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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