A295406 -id:A295406 - cp's OEIS frontend

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

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

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

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

Original entry on oeis.org

1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.

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 (m=1), A295385 (m=2), A295406 (m=3), A295407 (m=4).

Cf. A295409, A295418, A332679.

[Factorial(n)*(&+[Binomial(5*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,4*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[5*n,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(5*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

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

a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017

A332692 a(n) = n! * Laguerre(n, 2*n).

Original entry on oeis.org

1, -1, 2, 6, -232, 4120, -61488, 740432, -3220096, -224705664, 11713068800, -397487915264, 10466018491392, -176186211195904, -2178925657151488, 399827849856768000, -24748326426744881152, 1112888620945558700032, -36293785214959525625856, 408738923015995616067584

Offset: 0

Vaclav Kotesovec, Feb 20 2020

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A277423, A332693, A332679, A332694, A332695.

Cf. A277391, A295406.

Table[n! * LaguerreL[n, 2*n], {n, 0, 25}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * 2^k * n^k / k!, {k, 0, n}], {n, 1, 25}]}]
Table[n! * Hypergeometric1F1[-n, 1, 2*n], {n, 0, 25}]

a(n) = n!*pollaguerre(n, 0, 2*n); \\ Michel Marcus, Feb 05 2021

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

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

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

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A295408 a(n) = n! * Laguerre(n, 4*n, -n).

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A332692 a(n) = n! * Laguerre(n, 2*n).

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

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Magma

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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!.