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.

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A009283 E.g.f.: exp(x + sinh(x)).

Original entry on oeis.org

1, 2, 4, 9, 24, 73, 246, 913, 3688, 16057, 74954, 372749, 1965156, 10942285, 64103006, 393902353, 2532010800, 16982676561, 118600412626, 860680689429, 6478753957948, 50505684285301, 407133297257542, 3389160344023385, 29098108436107592, 257364794368638009
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A003724, A009227, A009282.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x+Sinh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 16 2022 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x + sinh(x)))) /* Joerg Arndt, Sep 01 2012 */

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997
Name corrected by Arkadiusz Wesolowski, Sep 01 2012

A009623 Expansion of sinh(x).exp(sinh(x)).

Original entry on oeis.org

0, 1, 2, 4, 12, 36, 118, 456, 1816, 7888, 37354, 184064, 974372, 5444544, 31769182, 195982208, 1259350576, 8441139456, 59073098706, 428299217920, 3226127944764, 25165446157312, 202778723085382, 1689266143553536
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

A003724, A009541.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Sinh[x]*Exp[Sinh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 07 2022 *)

Formula

a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A003724 and A009541. - Peter Bala, Dec 06 2011

Extensions

Extended and signs tested Mar 15 1997 by Olivier Gérard.
Previous Mathematica program replaced by Harvey P. Dale, Jun 07 2022

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).

A330021 Expansion of e.g.f. exp(sinh(exp(x) - 1)).

Original entry on oeis.org

1, 1, 2, 6, 25, 128, 754, 5001, 37048, 303930, 2732395, 26657106, 280039786, 3149224991, 37729906686, 479570263690, 6442902231289, 91186621152460, 1355582225366134, 21112253012491481, 343672026658191836, 5834977672879651390, 103130592695715620419
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

Comments

Stirling transform of A003724.
Exponential transform of A024429.

Links

Crossrefs

Cf. A003724, A009218, A011800, A024429, A080831.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*irem(j, 2)*g(n-j), j=1..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 23 2023
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sinh[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A003724(k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024429(k) * a(n-k).

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

A346746 E.g.f.: exp( (x * exp(x) - sinh(x)) / 2 ).

Original entry on oeis.org

1, 0, 1, 1, 5, 12, 58, 220, 1145, 5684, 33284, 198412, 1306355, 8945046, 65658392, 503505600, 4076565489, 34442610648, 304577372128, 2802673411280, 26840614943667, 266644080930194, 2745669007978680, 29243006731749200, 321810005123384617, 3653558357684804324
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Comments

Exponential transform of A004526.

Crossrefs

Cf. A000248, A003724, A004526, A346747, A346748.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[(x Exp[x] - Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Floor[k/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A004526(k) * a(n-k).

A346747 E.g.f.: exp( (x * exp(x) + sinh(x)) / 2 ).

Original entry on oeis.org

1, 1, 2, 6, 20, 79, 357, 1783, 9788, 58361, 374581, 2571851, 18779928, 145163975, 1183028095, 10129297307, 90843458256, 851083079649, 8309588493841, 84370700833147, 889152061199144, 9709123938880103, 109677977422359703, 1279880472867083111, 15408386793144717536
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Comments

Exponential transform of A110654.

Crossrefs

Cf. A000248, A003724, A110654, A346746, A346748.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[(x Exp[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Ceiling[k/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A110654(k) * a(n-k).

A346748 E.g.f.: exp( (x * exp(-x) + sinh(x)) / 2 ).

Original entry on oeis.org

1, 1, 0, 0, 4, -1, -9, 103, -132, -535, 7731, -25117, -18072, 1078215, -6917039, 16312667, 186611792, -2454241183, 14370311311, 1436259867, -934228834216, 10658996229479, -54990712418263, -185381404760729, 7270919988375200, -80130195880201583, 391992372213719679
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Comments

Exponential transform of A001057.

Crossrefs

Cf. A001057, A003724, A003725, A346746, A346747.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[(x Exp[-x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n - 1, k - 1] Floor[(k + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001057(k) * a(n-k).

Extensions

Typo in a(26) corrected by Georg Fischer, Nov 30 2021

A351891 Expansion of e.g.f. exp( sinh(sqrt(2)*x) / sqrt(2) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 25, 105, 443, 1969, 10609, 57265, 338547, 2190969, 14498185, 104277849, 784965803, 6150938593, 51229928929, 440694547681, 3967606065891, 37247506348905, 361022009762809, 3645855348771273, 38001754007842715, 409302848055407761, 4558828622414199121
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A055882, A136630, A351892, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 2^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 2^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025

A351892 Expansion of e.g.f. exp( sinh(sqrt(3)*x) / sqrt(3) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 40, 205, 952, 4921, 31168, 189145, 1318528, 9843781, 74869888, 632536933, 5475991552, 49996774897, 485393809408, 4829958877105, 50858117779456, 554544498995965, 6259096187060224, 73822470722135293, 894846287081242624, 11261265009125680681, 146272258394568687616
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A136630, A247452, A351891, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[3] x]/Sqrt[3]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 3^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 3^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025
Previous Showing 31-40 of 53 results. Next

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