A295408 -id:A295408 - cp's OEIS frontend

A332679 a(n) = (-1)^n * n! * Laguerre(n, 4*n).

1, 3, 34, 642, 16920, 571880, 23577552, 1147008912, 64304389504, 4081584090240, 289302692908800, 22648001532831488, 1940655970832219136, 180654087647513945088, 18153823412468554639360, 1958590905998560664832000, 225799980396482832660529152, 27702168947661388727726931968

Offset: 0

Vaclav Kotesovec, Feb 19 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..335

Cf. A277423, A332692, A332693, A332694, A332695.

Cf. A277418, A295408, A302112, A332680.

Table[(-1)^n*n!*LaguerreL[n, 4*n], {n, 0, 20}]
Join[{1}, Table[n! * Sum[(-1)^(n-k) * Binomial[n, k] * (4*n)^k/k!, {k, 0, n}], {n, 1, 20}]]
Table[(-1)^n*n!*Hypergeometric1F1[-n, 1, 4*n], {n, 0, 20}]

a(n) = (-1)^n*n!*pollaguerre(n, 0, 4*n); \\ Michel Marcus, Feb 05 2021

A302112(n) = (a(n) - 2*n*A332680(n)) * binomial(2*n, n) / 2^n.
a(n) / (n*A332680(n)) ~ 2.
a(n) ~ c * n^(n + 1/6) * exp(n), where c = Gamma(1/3) / (2^(5/6) * 3^(1/6) * sqrt(Pi)) = 0.706332637459...

A295385 a(n) = n!Sum_{k=0..n} binomial(2n,n-k)*n^k/k!.

Original entry on oeis.org

1, 3, 32, 579, 14736, 483115, 19376928, 918980139, 50306339072, 3121729082739, 216541483852800, 16603614676249843, 1394473165806440448, 127308860552307549531, 12553171419275174137856, 1329537514269062031406875, 150531055969843353812533248, 18143286205523964035258551651

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..335
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A082545, A277373, A277423, A295384, A295406, A295407, A295408, A295409.

[Factorial(n)*(&+[Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n! SeriesCoefficient[Exp[n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 17}]
Table[n! LaguerreL[n, n, -n], {n, 0, 17}]
Table[(-1)^n HypergeometricU[-n, n + 1, -n], {n, 0, 17}]
Join[{1}, Table[n! Sum[Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 17}]]

for(n=0,30, print1(n!*sum(k=0,n, binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n+1).
a(n) = n!*Laguerre(n,n,-n).
a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^n / exp((2 - sqrt(2))*n). - Vaclav Kotesovec, Nov 21 2017

A295406 a(n) = n! * Laguerre(n, 2*n, -n).

Original entry on oeis.org

1, 4, 58, 1422, 49000, 2174360, 118023264, 7574532826, 561071549056, 47111034709260, 4421715905632000, 458741213603157254, 52129735913348001792, 6439324687323193520608, 859089518697047400878080, 123108032319553206480143250, 18858657171509448248927617024

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295407, A295408.

[Factorial(n)*(&+[Binomial(3*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,2*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[3*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(3*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 2*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(3*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 5/(2*sqrt(13))) * ((11 + sqrt(13))/2)^n * exp((sqrt(13)-5)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(2*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295407 a(n) = n! * Laguerre(n, 3*n, -n).

Original entry on oeis.org

1, 5, 92, 2859, 124832, 7018105, 482598720, 39236322839, 3681751480832, 391611920476653, 46560370087846400, 6119025385880816035, 880818377346674454528, 137824220501484017301281, 23291983597732334528110592, 4228010378355969165140319375

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295408.

[Factorial(n)*(&+[Binomial(4*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,3*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[4*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(4*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 3*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(4*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 3/(2*sqrt(5))) * (8*(sqrt(5)-1))^n * exp((sqrt(5)-3)*n) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(3*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295409 a(n) = n! * Laguerre(n, n^2, -n).

Original entry on oeis.org

1, 3, 58, 2859, 267576, 40818095, 9235507968, 2906955312471, 1215257338052992, 651548571287972859, 435901423022852332800, 356000439852418418920643, 348583395952381998326141952, 403108990190536860168604229031, 543577365164816368801494214352896

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..214
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295407, A295408, A295418.

[Factorial(n)*(&+[Binomial(n*(n+1), n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 11 2018

seq(n!*orthopoly[L](n,n^2,-n),n=0..30); # Robert Israel, Nov 22 2017

Table[n!*LaguerreL[n,n^2,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n*(n+1),n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0,30, binomial(n*(n+1), n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 11 2018

a(n) = n!*pollaguerre(n, n^2, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n! * Sum_{k=0..n} binomial(n*(n+1),n-k)*n^k/k!.
a(n) ~ exp(3/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Nov 23 2017

A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

Original entry on oeis.org

1, 2, 32, 1422, 124832, 18246850, 4005713952, 1232956594814, 506672220394496, 267992015325604578, 177340024595660672000, 143531889358151618790862, 139482579412432078779322368, 160267575964062522718064075618, 214924620455826226723051817295872

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..213
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295407, A295408, A295409.

[Factorial(n)*(&+[Binomial(n^2, n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, May 13 2018

Table[n!*LaguerreL[n,n*(n-1),-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n^2,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,25, print1(n!*sum(k=0,n, binomial(n^2, n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 13 2018

a(n) = n!*pollaguerre(n, n*(n-1), -n); \\ Michel Marcus, Feb 05 2021

a(n) = n! * Sum_{k=0..n} binomial(n^2,n-k)*n^k/k!.
a(n) ~ exp(1/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2-n+1). - Ilya Gutkovskiy, Nov 23 2017

cp's OEIS Frontend

A332679 a(n) = (-1)^n * n! * Laguerre(n, 4*n).

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A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/k!.

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A295406 a(n) = n! * Laguerre(n, 2*n, -n).

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A295407 a(n) = n! * Laguerre(n, 3*n, -n).

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A295409 a(n) = n! * Laguerre(n, n^2, -n).

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A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

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A295385 a(n) = n!Sum_{k=0..n} binomial(2n,n-k)*n^k/k!.