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 21-28 of 28 results.

A336589 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(1 - exp(x))).

Original entry on oeis.org

1, 0, -3, -19, -75, 574, 25795, 579963, 9342529, 21955076, -7954085799, -535479422655, -25206613635203, -871888114433454, -7465407495946777, 2538884115164554199, 344689220434285963905, 31689538033223254172648, 2273498459548301881979029
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2020

Keywords

Crossrefs

Cf. A001569, A009940, A293037, A336588.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[1 - Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
Table[n! Sum[(-1)^k StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = n! * sum(k=0, n, (-1)^k*stirling(n+1,k+1,2) / k!); \\ Michel Marcus, Jul 29 2020

Formula

a(n) = n! * Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) / k!.

A336606 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) / BesselJ(0,2*sqrt(x)).

Original entry on oeis.org

1, 2, 9, 70, 851, 15246, 384147, 13065354, 578905875, 32440563766, 2243907466283, 187796863841346, 18704441632101337, 2186374265471576090, 296396762529435076953, 46126320892158605384334, 8167358455139620845210003, 1632571811017090501346518086
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2020

Keywords

Crossrefs

Cf. A000275, A002720, A009940, A336608.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[x]/BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[n_] := n! Sum[Binomial[n, k] A000275[k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 17}]

Formula

a(n) = n! * Sum_{k=0..n} binomial(n,k) * A000275(k) / k!.

A336608 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) / BesselJ(0,2*sqrt(x)).

Original entry on oeis.org

1, 0, 1, 4, 51, 856, 21435, 725796, 32132499, 1800176176, 124511280723, 10420458131260, 1037868062069113, 121317006426807192, 16446390218708245393, 2559445829942874207804, 453188354421968867989395, 90587738500599611033753184
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2020

Keywords

Crossrefs

Cf. A000275, A002720, A009940, A336606.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[-x]/BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[n_] := n! Sum[(-1)^(n - k) Binomial[n, k] A000275[k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 17}]

Formula

a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000275(k) / k!.

A336634 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) * BesselI(0,2*sqrt(x))^2.

Original entry on oeis.org

1, 1, 0, -4, 14, -18, -168, 1920, -11898, 27398, 582896, -13028904, 183020620, -2061910004, 17930433744, -65293856160, -1965585556410, 69343044999750, -1519055329884960, 26755366818127560, -374375460816570780, 2924763867241325220
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

Links

Crossrefs

Cf. A000984, A009940, A244973, A336293.

Programs

  • Maple
    rec:= n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=1,a(2)=0},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 30 2020
  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[-x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2
Table[(-1)^n n! HypergeometricPFQ[{1/2, -n}, {1, 1}, 4], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k)^2 * binomial(2*k,k) * (n-k)!); \\ Michel Marcus, Jul 30 2020

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(2*k,k) * (n-k)!.
D-finite with recurrence: n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3). - Robert Israel, Jul 30 2020

A346409 a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k)^2 * k!).

Original entry on oeis.org

0, 1, -3, 13, -52, 476, 1344, 156192, 6935424, 470168064, 38948065920, 3979380286080, 489922581219840, 71586095491054080, 12249193741572372480, 2426646293132502067200, 551096248249459158220800, 142236660450422499604070400, 41404182857569072540171468800
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Crossrefs

Cf. A002741, A009940, A082491, A142999, A346405.

Programs

  • Mathematica
    Table[(n!)^2 Sum[(-1)^k/((n - k)^2 k!), {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * exp(-x).

A352150 a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n,k)^2 * (n-k-1)!.

Original entry on oeis.org

0, 1, -3, 2, -6, -1, -5, 132, 1624, 17145, 174509, 1789842, 18659146, 196678143, 2057524963, 20460314396, 171030108768, 529697015489, -27050118799923, -1079945984126798, -30289996673371254, -765129844741436785, -18575997643525737477, -444653043972658034044
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2022

Keywords

Crossrefs

Cf. A002741, A009940, A352149.

Programs

  • Mathematica
    Table[Sum[(-1)^k Binomial[n, k]^2 (n - k - 1)!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; Assuming[x > 0, CoefficientList[Series[BesselJ[0, 2 Sqrt[x]] (ExpIntegralEi[x] - Log[x] - EulerGamma), {x, 0, nmax}], x]] Range[0, nmax]!^2
  • PARI
    a(n) = sum(k=0, n-1, (-1)^k * binomial(n,k)^2 * (n-k-1)!); \\ Michel Marcus, Mar 06 2022

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) * (Ei(x) - log(x) - gamma).

A356559 a(n) = exp(-1) * n! * Sum_{k>=0} Laguerre(n,k) / k!.

Original entry on oeis.org

1, 0, 0, 1, 7, 43, 281, 2056, 17004, 157809, 1622515, 18245335, 222004597, 2898508416, 40343356184, 595578837205, 9287308741827, 152459628788599, 2627373030049669, 47425289731038656, 895098852673047772, 17644305594671247141, 363065584549610882703, 7799894520723959486795
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2022

Keywords

Links

Crossrefs

Cf. A000110, A009940, A101053, A317362, A317366.

Programs

  • Mathematica
    Table[Exp[-1] n! Sum[LaguerreL[n, k]/k!, {k, 0, Infinity}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Exp[Exp[-x/(1 - x)] - 1]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k]^2 k! BellB[n - k], {k, 0, n}], {n, 0, 23}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(-x/(1 - x)) - 1) / (1 - x))) \\ Michel Marcus, Aug 12 2022

Formula

E.g.f.: exp(exp(-x/(1 - x)) - 1) / (1 - x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * k! * Bell(n-k).

A371114 a(n) is n! times the coefficient of x in the associated Laguerre polynomial Laguerre(n,x,1).

Original entry on oeis.org

0, 1, 1, -1, -16, -111, -691, -4145, -23080, -96159, 195137, 13914911, 284958904, 4842967921, 77613841629, 1219132694767, 19006810258064, 294117291312577, 4469910552829473, 64942899785556031, 841752172982238304, 7465153745073705041, -61832090598783228403, -6408471053640082778097
Offset: 0

Views

Author

Rui Xian Siew, Mar 10 2024

Keywords

Crossrefs

Cf. A009940.

Programs

  • Mathematica
    a[n_]:=n! D[LaguerreL[n,x,1],x]/.{x->0}; Array[a,25,0]
Table[n! Sum[LaguerreL[k,1]/(n-k),{k,0,n-1}],{n,0,25}]
RecurrenceTable[{(-3 + n)^3*(-2 + n)*a[n-4] - (-2 + n)*(28 - 21*n + 4*n^2)*a[n-3] + (27 - 25*n + 6*n^2)*a[n-2] + (7 - 4*n)*a[n-1] + a[n] == 0, a[0] == 0, a[1] == 1, a[2] == 1, a[3] == -1}, a, {n, 0, 20}] (* Vaclav Kotesovec, Mar 12 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n-1, pollaguerre(k, 0, 1)/(n-k)); \\ Michel Marcus, Mar 12 2024

Formula

a(n) = n!*Sum_{k=0..n-1} A009940(k)/(k!*(n-k)).
a(n) = (4*n - 7)*a(n-1) - (6*n^2 - 25*n + 27)*a(n-2) + (n-2)*(4*n^2 - 21*n + 28)*a(n-3) - (n-3)^3*(n-2)*a(n-4). - Vaclav Kotesovec, Mar 12 2024
Previous Showing 21-28 of 28 results.

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