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 1-4 of 4 results.

A087912 Exponential generating function is exp(2*x/(1-x))/(1-x).

Original entry on oeis.org

1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856, 1815177984, 29808908032, 525586164736, 9898343691264, 198227905206272, 4204989697906688, 94163381359509504, 2219240984918720512, 54898699229094412288, 1422015190821016633344, 38484192401958599131136
Offset: 0

Views

Author

Vladeta Jovovic, Oct 18 2003

Keywords

Links

Crossrefs

Cf. A002720, A274246, A277372.
Column k=2 of A289192.
Cf. A160615, A160616.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 16 2018
  • Maple
    a := proc(n) option remember: if n<1 then 1 else (2*n+1)*a(n-1) - (n-1)^2*a(n-2) fi end: 'a(n)'$n=0..17; # Zerinvary Lajos, Sep 26 2006; corrected by M. F. Hasler, Sep 30 2012
  • Mathematica
    Table[n! SeriesCoefficient[E^(2*x/(1-x))/(1-x), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)
Table[n!*LaguerreL[n, -2], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)
  • PARI
    A087912(n)={n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,2^m*x^m/m!^2) ,n)} \\ Paul D. Hanna, Nov 18 2011
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(2*x/(1-x))/(1-x))) \\ Joerg Arndt, May 10 2013

Formula

E.g.f.: exp(2*x/(1-x))/(1-x). - M. F. Hasler, Sep 30 2012
a(n) = n!*LaguerreL(n, -2).
a(n) = Sum_{k=0..n} 2^k*(n-k)!*binomial(n, k)^2.
E.g.f.: exp(x) * Sum_{n>=0} 2^n*x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]
a(n) ~ n^(n+1/4)*exp(2*sqrt(2*n)-n-1)*2^(-3/4). - Vaclav Kotesovec, Sep 29 2012
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(2*n))) = exp(-1). - Vaclav Kotesovec, Oct 12 2016
a(n) = n! * A160615(n)/A160616(n). - Alois P. Heinz, Jun 28 2017
D-finite with recurrence: a(n) +(-2*n-1)*a(n-1) +(n-1)^2*a(n-2)=0. - R. J. Mathar, Feb 21 2020

Extensions

Several minor edits by M. F. Hasler, Sep 30 2012

A277386 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.

Original entry on oeis.org

1, 4, 35, 438, 6873, 127488, 2703447, 64121130, 1674999009, 47638235484, 1461975938379, 48068355965886, 1683311251028265, 62477888170824792, 2447583053876363727, 100842325515959413842, 4356021203508275392833, 196739133595421931988020, 9268144156277932321747251
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 12 2016

Keywords

Crossrefs

Cf. A000172, A277382, A241247, A274246.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^3 * 3^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

Formula

Recurrence: n*(8*n - 23)*a(n) = 3*(8*n^3 - 15*n^2 - 30*n + 17)*a(n-1) - (n-1)*(24*n^3 - 261*n^2 + 770*n - 666)*a(n-2) + (n-2)^3*(n-1)*(8*n - 15)*a(n-3).
a(n) ~ n^(n - 1/6) * exp(3*3^(1/3)*n^(2/3) - 3^(2/3)*n^(1/3) - n +1) / (3^(5/6)*sqrt(2*Pi)) * (1 + 19/(6*3^(2/3)*n^(1/3)) + 1193/(1080*3^(1/3) * n^(2/3))).
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 3^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

A354941 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-2)^(n-k).

Original entry on oeis.org

1, -1, -10, -2, 488, 4088, -9968, -730480, -9751936, -11540096, 2480655104, 62522038016, 680469314560, -8292439149568, -606011029669888, -15765339965278208, -183530875864317952, 4164677242501038080, 318357069130977181696, 10359690304436314505216, 176911847384965046337536
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 12 2022

Keywords

Crossrefs

Cf. A000023, A274246, A295382, A343840, A354942.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^3 k! (-2)^(n - k), {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[BesselI[0, 2 Sqrt[x]] Sum[(-2)^k x^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-2)^(n-k)); \\ Michel Marcus, Jun 12 2022

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} (-2)^n * x^n / n!^3.

A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

Original entry on oeis.org

1, 2, 22, 438, 12824, 496370, 23914512, 1379269094, 92667551104, 7108231236066, 612974464428800, 58702772664490262, 6181602019316333568, 709911177607125141362, 88301595129435811723264, 11825985945777638231211750, 1696696168760520436580974592, 259624546758869333450285984066
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 12 2022

Keywords

Crossrefs

Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^3 * k! * n^(n-k)); \\ Michel Marcus, Jun 12 2022

Formula

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.
a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - Vaclav Kotesovec, Jul 01 2022, updated Mar 17 2024
Showing 1-4 of 4 results.

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