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

A160607 Numerator of Laguerre(n, -6).

Original entry on oeis.org

1, 7, 31, 109, 331, 4529, 11453, 190433, 61391, 928943, 9677971, 214858067, 60236303, 10492980947, 137504412401, 180206776249, 2272545257401, 68446106098751, 119418042814439, 3899046884359, 33130663362484669
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For denominators see A160608.
Cf. A289212.

Programs

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