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 1-3 of 3 results.

A289212 a(n) = n! * Laguerre(n,-6).

Original entry on oeis.org

1, 7, 62, 654, 7944, 108696, 1649232, 27422352, 495057024, 9631281024, 200682406656, 4455296877312, 104921038236672, 2610989435003904, 68430995893131264, 1883330926998829056, 54286270223002140672, 1635031821385383247872, 51347572582353094508544
Offset: 0

Views

Author

Alois P. Heinz, Jun 28 2017

Keywords

Links

Crossrefs

Column k=6 of A289192.
Cf. A160607, A160608.

Programs

  • Maple
    a:= n-> n! * add(binomial(n, i)*6^i/i!, i=0..n):
seq(a(n), n=0..20);
  • Mathematica
    Table[n!*LaguerreL[n, -6], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)
  • PARI
    my(x = 'x + O('x^30)); Vec(serlaplace(exp(6*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017
  • PARI
    a(n) = n!*pollaguerre(n, 0, -6); \\ Michel Marcus, Feb 05 2021
  • Python
    from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -6))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(6*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 6^i/i! * binomial(n,i).
a(n) = n! * A160607(n)/A160608(n).
a(n) ~ exp(-3 + 2*sqrt(6*n) - n) * n^(n + 1/4) / (2^(3/4)*3^(1/4)) * (1 + 97/(16*sqrt(6*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 6^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A160608 Denominator of Laguerre(n, -6).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 35, 5, 35, 175, 1925, 275, 25025, 175175, 125125, 875875, 14889875, 14889875, 282625, 1414538125, 9901766875, 2222845625, 2505147019375, 2505147019375, 62628675484375, 116310397328125, 814172781296875
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For numerators see A160607.
Cf. A289212.

Programs

  • Magma
    [Denominator((&+[Binomial(n,k)*(6^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 12 2018
  • Mathematica
    Denominator[Table[LaguerreL[n, -6], {n, 0, 50}]] (* G. C. Greubel, May 12 2018 *)
  • PARI
    for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*(6^k/k!))), ", ")) \\ G. C. Greubel, May 12 2018
  • PARI
    a(n) = denominator(pollaguerre(n, 0, -6)); \\ Michel Marcus, Feb 05 2021

A289213 a(n) = n! * Laguerre(n,-7).

Original entry on oeis.org

1, 8, 79, 916, 12113, 179152, 2921911, 51988748, 1000578817, 20686611736, 456805020959, 10721879413252, 266382974861521, 6980304560060384, 192311632290456007, 5555079068684580988, 167822887344661475969, 5290815252203206305832, 173713426149927498289903
Offset: 0

Views

Author

Alois P. Heinz, Jun 28 2017

Keywords

Links

Crossrefs

Column k=7 of A289192.
Cf. A160607, A160608.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(7*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 13 2018
  • Maple
    a:= n-> n! * add(binomial(n, i)*7^i/i!, i=0..n):
seq(a(n), n=0..20);
  • Mathematica
    Table[n!*LaguerreL[n, -7], {n, 0, 20}] (* Indranil Ghosh, Jul 04 2017 *)
  • PARI
    x = 'x + O('x^30); Vec(serlaplace(exp(7*x/(1-x))/(1-x))) \\ Michel Marcus, Jul 04 2017
  • Python
    from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -7))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

Formula

E.g.f.: exp(7*x/(1-x))/(1-x).
a(n) = n! * Sum_{i=0..n} 7^i/i! * binomial(n,i).
a(n) = n! * A160607(n)/A160608(n).
a(n) ~ exp(-7/2 + 2*sqrt(7*n) - n) * n^(n + 1/4) / (sqrt(2)*7^(1/4)) * (1 + 367/(48*sqrt(7*n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 7^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020
Showing 1-3 of 3 results.

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

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