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.

A119400 a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n,k).

Original entry on oeis.org

1, 2, 13, 172, 3809, 126526, 5874517, 362848088, 28744087297, 2839192902874, 341922922464701, 49297062811573732, 8380916229314577313, 1658770724530766046422, 378056469777362366873989, 98286603829297813268996176, 28907477297195536067142301697
Offset: 0

Views

Author

Vladeta Jovovic, Jul 25 2006

Keywords

Links

Crossrefs

Cf. A000522, A002720, A216831.

Programs

  • Mathematica
    Table[Sum[(n!/k!)^2*Binomial[n, k], {k, 0, n}], {n, 0, 16}] (* Stefan Steinerberger, Jun 17 2007 *)
  • PARI
    a(n)=n!^3*polcoeff(exp(x+x*O(x^n))*sum(m=0, n, x^m/m!^3), n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 27 2012

Formula

Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1-x)))/(1-x).
Recurrence: a(n) = (3*n^2-3*n+2)*a(n-1)-3*(n-1)^4*a(n-2)+(n-2)^3*(n-1)^3*a(n-3). - Vaclav Kotesovec, Sep 30 2012
a(n) ~ 1/sqrt(3)*n^(2*n+2/3)/exp(2*n-3*n^(1/3)). - Vaclav Kotesovec, Sep 30 2012
E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^3 = Sum_{n>=0} a(n)*x^n/n!^3. - Paul D. Hanna, Nov 27 2012
a(n) = Sum_{k=0..n} k!^2*binomial(n,k)^3. - Ridouane Oudra, Jun 14 2025

Extensions

More terms from Stefan Steinerberger, Jun 17 2007

A274246 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 2^(n-k) * k!.

Original entry on oeis.org

1, 3, 22, 230, 3048, 48152, 875536, 17907024, 405320320, 10030449536, 268836428544, 7744939895552, 238352004594688, 7795463142466560, 269761049981827072, 9839883848966985728, 377091995258812268544, 15139047281589466136576, 635088889901946682408960, 27775758544209632635060224, 1263876454164193257295446016
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 12 2016

Keywords

Links

Crossrefs

Cf. A000172, A087912, A216831, A241247, A277386.

Programs

  • Magma
    [(&+[Binomial(n,j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Dec 27 2022
  • Maple
    f:= gfun:-rectoproc({n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3),a(0)=1,a(1)=3,a(2)=22},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 16 2017
  • Mathematica
    Table[Sum[Binomial[n, k]^3 * 2^(n-k) * k!, {k, 0, n}], {n, 0, 20}]
  • SageMath
    def A274246(n): return sum(binomial(n,j)^3*factorial(j)*2^(n-j) for j in range(n+1))
[A274246(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

Formula

Recurrence: n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3).
a(n) ~ n^(n - 1/6) * exp(3*2^(1/3)*n^(2/3) - 2^(2/3)*n^(1/3) - n + 2/3) / (2^(5/6)*sqrt(3*Pi)) * (1 + 31*2^(1/3)/(27*n^(1/3)) + 3437/(3645*2^(1/3) * n^(2/3))).
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 2^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022
a(n) = 2^n * Hypergeometric3F1([-n, -n, -n], [1], -1/2). - G. C. Greubel, Dec 27 2022

A343840 a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(n, k)*|A021009(n, k)|.

Original entry on oeis.org

1, 0, -5, 22, 9, -1244, 14335, -79470, -586943, 25131304, -434574909, 4418399470, 8524321465, -1771817986548, 53502570125719, -1052208254769014, 11804172888840705, 131741085049224400, -12970386000411511733, 482732550618027365574, -12599999790172579025879
Offset: 0

Views

Author

Peter Luschny, May 04 2021

Keywords

Comments

Related to the coefficient triangle of generalized Laguerre polynomials A021009.

Crossrefs

Cf. A021009, A216831.

Programs

  • Maple
    T := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
a := n -> add((-1)^(n-j)*T(n, j)*binomial(n, j), j=0..n): seq(a(n), n=0..20);
  • PARI
    rowT(n) = Vecrev(n!*pollaguerre(n)); \\ A021009
a(n) = my(v=rowT(n)); sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(v[k+1])); \\ Michel Marcus, May 04 2021

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = BesselJ(0,2*sqrt(x)) * Sum_{n>=0} x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022
a(n) = Sum_{k=0..n} (-1)^k*k!*binomial(n,k)^3. - Ridouane Oudra, Jul 11 2025
Recurrence: n*(8*n - 11)*a(n) = -(n-1)*(24*n^2 - 49*n + 21)*a(n-1) - (n-1)*(24*n^3 - 33*n^2 - 14*n + 18)*a(n-2) - (n-2)^3*(n-1)*(8*n - 3)*a(n-3). - Vaclav Kotesovec, Jul 11 2025

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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