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.

A295256 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cosh(x))).

Original entry on oeis.org

1, 1, 4, 33, 384, 5945, 115680, 2713417, 74568704, 2350925649, 83660474880, 3317599815761, 145087264278528, 6937450761100873, 360078818344534016, 20162761727269502265, 1211588127198611374080, 77769423447774393465377, 5310706204624302598127616, 384439720034220718046773249
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A185951, A295237, A295254, A295255, A295257, A295258.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*cosh(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 Cosh[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Cosh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2 + 2*r*sqrt(1-16*r^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.2428073624074744554637516823... is the root of the equation 2*r*(exp(2*r)+1) = exp(r). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A185951(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

A381428 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 73, 1344, 33481, 1054656, 40223233, 1802385024, 92827015921, 5403527705856, 350854589607193, 25142008355656704, 1971003462240791161, 167802783944207917056, 15417877986778302551953, 1520661128893781018640384, 160249491538400609431567201, 17969682580669053325124960256
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Crossrefs

Cf. A006154, A295254, A381429.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*k+1,k)/(3*k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(3*k+1,k)/(3*k+1) * A136630(n,k).

A381429 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x) * A(x)^3 ).

Original entry on oeis.org

1, 1, 8, 133, 3392, 117601, 5167808, 275334613, 17250670592, 1242994578721, 101273185092608, 9206681997173893, 923928346115182592, 101453787213382443841, 12100018549609932996608, 1557645163271323384461973, 215265839194368088629051392, 31788685348087376561935104961
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Crossrefs

Cf. A006154, A295254, A381428.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(4*k+1, k)/(4*k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(4*k+1,k)/(4*k+1) * A136630(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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