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.

A277373 a(n) = Sum_{k=0..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

Original entry on oeis.org

1, 2, 14, 168, 2840, 61870, 1649232, 51988748, 1891712384, 78031713690, 3598075308800, 183396819358192, 10239159335648256, 621414669926828102, 40733145577028065280, 2867932866586451980500, 215859025837098699948032, 17295664826665032427023922, 1469838791737283957748596736
Offset: 0

Views

Author

Peter Luschny, Oct 12 2016

Keywords

Comments

Limit_{n -> infinity} (LaguerreL(n,-n)/BesselI(0,2*n))^(1/n) = exp(-2 + 1/phi) * phi^2 = 0.657347578792874..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2016
For m > 0, n!*LaguerreL(n, -m*n) ~ sqrt(1/2 + (m+2)/(2*sqrt(m*(m+4)))) * (2+m+sqrt(m*(m+4)))^n * exp(n*(sqrt(m*(m+4))-m-2)/2) * n^n / 2^n. - Vaclav Kotesovec, Oct 14 2016
For m > 4, (-1)^n * n! * LaguerreL(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n. - Vaclav Kotesovec, Feb 20 2020

Links

Crossrefs

Cf. A002720 (n!L(n,-1)), A087912 (n!L(n,-2)), A277382 (n!L(n,-3)), A277372 (n!L(n,-n)-n^n), A277423 (n!L(n,n)), A144084 (polynomials).
Cf. A277391 (n!L(n,-2*n)), A277392 (n!L(n,-3*n)), A277418 (n!L(n,-4*n)), A277419 (n!L(n,-5*n)), A277420 (n!L(n,-6*n)), A277421 (n!L(n,-7*n)), A277422 (n!L(n,-8*n)).
Main diagonal of A289192.

Programs

  • Magma
    [(&+[Binomial(n, n-k)*Binomial(n, k)*n^(n-k)*Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 16 2018
  • Maple
    A277373 := n -> n!*LaguerreL(n, -n): seq(simplify(A277373(n)), n=0..18);
# second Maple program:
a:= n-> n! * add(binomial(n, i)*n^i/i!, i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 27 2017
  • Mathematica
    Table[n!*LaguerreL[n, -n], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)
  • PARI
    a(n) = sum(k=0,n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!) \\ Charles R Greathouse IV, Feb 07 2017
  • PARI
    a(n) = n!*pollaguerre(n, 0, -n); \\ Michel Marcus, Feb 05 2021
  • Sage
    @cached_function
def L(n, x):
    if n == 0: return 1
    if n == 1: return 1 - x
    return (L(n-1,x) * (2*n-1-x) - L(n-2,x)*(n-1))/n
A277373 = lambda n: factorial(n)*L(n, -n)
print([A277373(n) for n in (0..20)])

Formula

a(n) = p(n,n) where p(n,x) = Sum_{k=0..n} binomial(n,n-k)*x^(n-k)*n!/(n-k)!. The coefficients of these polynomials are in A144084 (sorted by falling powers).
a(n) = n!*LaguerreL(n, -n).
a(n) = (-1)^n*KummerU(-n, 1, -n).
a(n) = n^n*hypergeom([-n, -n], [], 1/n) for n>=1.
a(n) ~ n^n * phi^(2*n+1) * exp(n/phi-n) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2016
a(n) = n! * [x^n] exp(n*x/(1-x))/(1-x). - Alois P. Heinz, Jun 28 2017
a(n) = n!^2 * [x^n] exp(x) * BesselI(0,2*sqrt(n*x)). - Ilya Gutkovskiy, Jun 19 2022

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

A277382 a(n) = n!*LaguerreL(n, -3).

Original entry on oeis.org

1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164, 8219695239, 148262469336, 2860241078817, 58736954622492, 1278727896354687, 29406849577341552, 712119108949808193, 18108134430393657636, 482306685868464422391, 13425231879291031821576
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 12 2016

Keywords

Comments

For m > 0, n!*LaguerreL(n, -m) ~ exp(2*sqrt(m*n) - n - m/2) * n^(n + 1/4) / (sqrt(2)*m^(1/4)) * (1 + (3+24*m+4*m^2)/(48*sqrt(m*n))).

Links

Crossrefs

Cf. A002720, A087912, A277372.
Column k=3 of A289192.
Cf. A160613, A160614.

Programs

  • Magma
    [Factorial(n)*((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018
  • Mathematica
    Table[n!*LaguerreL[n, -3], {n, 0, 20}]
CoefficientList[Series[E^(3*x/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]^2 * 3^k * (n-k)!, {k,0,n}], {n, 0, 20}]
  • PARI
    for(n=0,30, print1(n!*(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 09 2018

Formula

E.g.f.: exp(3*x/(1-x))/(1-x).
a(n) = Sum_{k=0..n} 3^k*(n-k)!*binomial(n, k)^2.
a(n) ~ exp(2*sqrt(3*n)-n-3/2) * n^(n+1/4) / (sqrt(2) * 3^(1/4)) * (1 + 37/(16*sqrt(3*n))).
D-finite with recurrence a(n) = 2*(n+1)*a(n-1) - (n-1)^2*a(n-2).
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(3*n))) = exp(-3/2).
a(n) = n! * A160613(n)/A160614(n). - Alois P. Heinz, Jun 28 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 3^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

Original entry on oeis.org

1, 1, 8, 99, 1696, 37225, 997056, 31535371, 1150303232, 47538819729, 2195314048000, 112032721984051, 6261138045038592, 380309520560089081, 24946892219825709056, 1757549042234670166875, 132356128415391650676736, 10610067001068927596601889, 902057202129607760380428288
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2017

Keywords

Links

Crossrefs

Main diagonal of A253286.
Cf. A000262, A052857, A052897, A255806, A277372.

Programs

  • Magma
    [n eq 0 select 1 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 1), -n): n in [0..20]]; // G. C. Greubel, Feb 23 2021
  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x/(1 - x)], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Product[Exp[n x^k], {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[n^k n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 18}]]
Join[{1}, Table[n n! Hypergeometric1F1[1 - n, 2, -n], {n, 1, 18}]]
Table[If[n==0, 1, n!*LaguerreL[n-1, 1, -n]], {n, 0, 20}] (* G. C. Greubel, Feb 23 2021 *)
  • PARI
    {a(n) = if(n==0, 1, n!*sum(k=1, n, n^k*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021
  • PARI
    a(n) = if (n, n! * pollaguerre(n-1, 1, -n), 1); \\ Michel Marcus, Feb 23 2021
  • Sage
    [1 if n==0 else factorial(n)*gen_laguerre(n-1, 1, -n) for n in (0..20)] # G. C. Greubel, Feb 23 2021

Formula

a(n) = n! * [x^n] Product_{k>=1} exp(n*x^k).
a(n) ~ exp(n/phi - n) * phi^(2*n) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 01 2017
a(n) = n! * Sum_{k=1..n} n^k * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021
a(n) = n! * LaguerreL(n-1, 1, -n) with a(0) = 1. - G. C. Greubel, Feb 23 2021
Showing 1-4 of 4 results.

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