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.

Previous Showing 11-14 of 14 results.

A295384 a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(2*n,n-k)*n^k/k!.

Original entry on oeis.org

1, 1, 0, -15, -112, -135, 9504, 152425, 610560, -27692847, -765107200, -6289891839, 213472972800, 9380264146825, 129378550468608, -3294028613874375, -226623617585053696, -4707649131227927775, 83803818828756418560, 9446689798312021406353, 277055229100887244800000
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

Links

Crossrefs

Cf. A006902, A277373, A277423, A295385.

Programs

  • Magma
    [Factorial(n)*(&+[(-1)^k*Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018
  • Maple
    a := n -> pochhammer(n, n)*hypergeom([1 - n], [n], n):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023
  • Mathematica
    Table[n! SeriesCoefficient[Exp[-n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 20}]
Table[n! LaguerreL[n, n, n], {n, 0, 20}]
Table[(-1)^n HypergeometricU[-n, n + 1, n], {n, 0, 20}]
Join[{1}, Table[n! Sum[(-1)^k Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 20}]]
  • PARI
    for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

Formula

a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x)^(n+1).
a(n) = n!*Laguerre(n,n,n).
a(n) = Pochhammer(n, n)*hypergeom([1 - n], [n], n). - Peter Luschny, Mar 23 2023

A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

Original entry on oeis.org

1, 3, 41, 1531, 111393, 13262051, 2336744233, 570621092091, 184341785557121, 76092709735150723, 39064090158380196201, 24408768326642565035963, 18237590837527919131499041, 16056004231253610384348995811, 16448689708899063469247204152553
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 16 2016

Keywords

Crossrefs

Cf. A086331, A277373, A277423, A277452.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n, k]*2^k*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

Formula

a(n) = exp(1/(2*n)) * 2^n * n^n * Gamma(n+1, 1/(2*n)).
a(n) ~ 2^n * n^n * n!.

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2019

Keywords

Crossrefs

Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

Programs

  • Magma
    [Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019
  • Mathematica
    Join[{1}, Table[n! Sum[(-1)^k Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, 1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[-x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = n! * [x^n] exp(-x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselJ(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A354944 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-n)^(n-k).

Original entry on oeis.org

1, 0, -10, 60, 1560, -39880, -491760, 45672060, -155935360, -77656158000, 2116774828800, 166585352850620, -11925674437248000, -330617542587341880, 69148933431781898240, -543549949643024194500, -434534462104188331130880, 21521903478880966780355360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 12 2022

Keywords

Crossrefs

Cf. A134095, A277423, A336180, A343840, A354941, A354942, A354943.

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.
Previous Showing 11-14 of 14 results.

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

Content is available under The OEIS End-User License Agreement.