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.

Previous Showing 51-58 of 58 results.

A319360 Expansion of e.g.f. (1 + x)*exp(log(1 + x)^2/2).

Original entry on oeis.org

1, 1, 1, 0, 2, -10, 64, -476, 4038, -38466, 406446, -4716624, 59621748, -815339460, 11992028112, -188746844040, 3165161922492, -56333871521508, 1060525150393308, -21053827255670976, 439558554065307288, -9627439778044075512, 220722057792327097920, -5286159770781782374800
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

Comments

Inverse Stirling transform of A000085.

Links

Crossrefs

Cf. A000085, A004211.

Programs

  • Maple
    seq(n!*coeff(series((1 + x)*exp(log(1 + x)^2/2),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[(1 + x) Exp[Log[1 + x]^2/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] HypergeometricU[-k/2, 1/2, -1/2]/(-1/2)^(k/2), {k, 0, n}], {n, 0, 23}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000085(k).

A369784 Expansion of e.g.f. exp( (exp(2*(exp(x)-1))-1)/2 ).

Original entry on oeis.org

1, 1, 4, 21, 137, 1068, 9663, 99249, 1137858, 14373531, 198031153, 2951536030, 47270242621, 808917666365, 14720125466652, 283667520561633, 5768057979319853, 123364873473674732, 2767400573883314755, 64950007415991458989, 1591227433994704322322
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Crossrefs

Cf. A000258, A369785.
Cf. A000085, A004211, A346417.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(2*(exp(x)-1))-1)/2)))

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A004211(k).

A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/2 ).

Original entry on oeis.org

1, 1, 7, 63, 713, 9753, 156111, 2858103, 58845105, 1344371793, 33713484151, 919838859151, 27105053988793, 857310780134825, 28953291147179007, 1039373409620929671, 39505610599553955809, 1584411299793530257697, 66846625774893448843879
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2024

Keywords

Links

Crossrefs

Cf. A080893, A375173.
Cf. A004211.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp((exp(4*x)-1)/2)-1)/2)))

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * Stirling2(n,k) * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * Stirling2(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2025

Keywords

Links

Crossrefs

Cf. A049376, A375173, A380257.
Cf. A004211, A025168.

Programs

  • Mathematica
    CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

Formula

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/2)) * (-5)^n * n! * Sum_{k>=0} binomial(-2*k/5,n)/(2^k * k!).

A380262 Expansion of e.g.f. exp( ((1+5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, -2, 16, -206, 3682, -84236, 2348704, -77241380, 2926735516, -125540336024, 6013069027648, -318093606114536, 18418565715581656, -1158626159228481488, 78679416565851286144, -5736477278907382585328, 446936684375920051751440, -37056888825921886749507872
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2025

Keywords

Links

Crossrefs

Cf. A000085, A002119, A380261.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(((1+5*x)^(2/5)-1)/2)))

Formula

a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/2)) * 5^n * n! * Sum_{k>=0} binomial(2*k/5,n)/(2^k * k!).

A305710 Expansion of e.g.f. exp(sec(x)*exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 11, 53, 297, 1959, 14499, 120409, 1097025, 10931771, 117685163, 1363889133, 16887554569, 222672557631, 3110742121059, 45912214062961, 713290136581697, 11636755988405555, 198800967493444875, 3549276499518132325, 66076184834921382313, 1280502976522048458647
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2018

Keywords

Examples

			exp(sec(x)*exp(x) - 1) = 1 + x + 3*x^2/2! + 11*x^3/3! + 53*x^4/4! + 297*x^5/5! + 1959*x^6/6! + 14499*x^7/7! + ...

Crossrefs

Cf. A000364, A003701, A004211, A009006, A009212, A009248, A009268, A305708.

Programs

  • Maple
    a:=series(exp(sec(x)*exp(x)-1),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sec[x] Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(2 I)^k EulerE[k, 1/2 - I/2] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

A323630 Expansion of e.g.f. exp(log(1 - x)^2/2)/(1 - x). This is also the transform of the involution numbers given by the signless Stirling cycle numbers.

Original entry on oeis.org

1, 1, 3, 12, 62, 390, 2884, 24472, 234086, 2490030, 29139306, 371878056, 5138306700, 76398336924, 1215973642584, 20624305367520, 371309259462972, 7071037633297116, 141997246553420052, 2998654325698019280, 66426777891686458728, 1540117294435707244488, 37296711627004301923056
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2019

Keywords

Crossrefs

Cf. A000085, A004211, A008275, A130534, A319360.

Programs

  • Maple
    seq(n!*coeff(series(exp(log(1-x)^2/2)/(1-x),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HypergeometricU[-k/2, 1/2, -1/2]/(-1/2)^(k/2), {k, 0, n}], {n, 0, 22}]
  • PARI
    my(x='x + O('x^25)); Vec(serlaplace(exp(log(1 - x)^2/2)/(1 - x))) \\ Michel Marcus, Jan 24 2019

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000085(k).
From Emanuele Munarini, Jul 09 2022: (Start)
a(n) = Sum_{k=0..n/2} |Stirling1(n+1,2*k+1)|*binomial(2*k,k)*k!/2^k.
a(n+1) = (n+1)*a(n) - Sum_{k=1..n} binomial(n,k)*(k-1)!*a(n-k). (End)

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).

Original entry on oeis.org

1, 2, 11, 92, 1025, 14232, 236403, 4568720, 100670529, 2490511776, 68341981051, 2059882505408, 67645498798721, 2403948686290816, 91914992104815459, 3762299973887526144, 164148252324092964993, 7604537914425558921728, 372812121514187124192875
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

Crossrefs

Cf. A004211, A007405, A134980, A337010, A337011.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).
a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).
Previous Showing 51-58 of 58 results.

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

Content is available under The OEIS End-User License Agreement.