A277423 -id:A277423 - cp's OEIS frontend

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

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

Offset: 0

Peter Luschny, Oct 12 2016

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

Alois P. Heinz, Table of n, a(n) for n = 0..356
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind
Wikipedia, Laguerre polynomials

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.

[(&+[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

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

Table[n!*LaguerreL[n, -n], {n, 0, 30}] (* G. C. Greubel, May 16 2018 *)

a(n) = sum(k=0,n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!) \\ Charles R Greathouse IV, Feb 07 2017

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

@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)])

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

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

Original entry on oeis.org

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...

A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

Original entry on oeis.org

1, 2, 13, 226, 7889, 458026, 39684637, 4788052298, 766526598721, 157108817646514, 40104442275129101, 12472587843118746322, 4641978487740740993233, 2036813028164774540010266, 1040451608604560812273060189, 612098707457003526384666111226

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..232

Main diagonal of A320031.

Cf. A086331, A277373, A277423, A277453.

a := n -> simplify(hypergeom([1, -n], [], -n)):
seq(a(n), n=0..15); # Peter Luschny, Oct 03 2018
# second Maple program:
b:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*b(n-1, k), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 09 2020

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

a(n) = sum(k=0, n, binomial(n,k) * n^k * k!); \\ Michel Marcus, Sep 18 2018

a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).
a(n) ~ n^n * n!.
a(n) = n! * [x^n] exp(x)/(1 - n*x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = floor(n^n*n!*exp(1/n)), n > 0. - Peter McNair, Dec 20 2021

A295382 Expansion of e.g.f. exp(-2*x/(1 - x))/(1 - x).

Original entry on oeis.org

1, -1, -2, -2, 8, 88, 592, 3344, 14464, 2944, -1121536, -21603584, -317969408, -4202380288, -51322677248, -562045749248, -4751724347392, -3419742961664, 1260396818661376, 45221885372727296, 1218206507254153216, 29421299633821057024, 669044215287581769728, 14528992234596624498688

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

sign

G. C. Greubel, Table of n, a(n) for n = 0..449
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=2 of A295381.

Cf. A009940, A087912, A160623, A160624, A277423.

[Factorial(n)*(&+[(-1)^k*Binomial(n,k)*2^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

a:=series(exp(-2*x/(1-x))/(1-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Exp[-2 x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! LaguerreL[n, 2], {n, 0, 23}]
Table[n! Hypergeometric1F1[-n, 1, 2], {n, 0, 23}]
Table[n! Sum[(-1)^k Binomial[n, k] 2^k/k!, {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(-2*x/(1-x))/(1-x))) \\ G. C. Greubel, Feb 06 2018

E.g.f.: exp(-2*x/(1 - x))/(1 - x).
a(n) = n!*Laguerre(n,2).
a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(n,k)*2^k/k!.
a(n) = n!*A160623(n)/A160624(n).
a(n) = Sum_{k=0..n} (-2)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 08 2025

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

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

A332693 a(n) = n! * Laguerre(n, 3*n).

Original entry on oeis.org

1, -2, 14, -156, 2328, -42630, 902736, -20961864, 497925504, -10347816906, 54902188800, 15803663268492, -1741565563831296, 146556727320337074, -11551833579195721728, 901051402625901468000, -71007771313742983888896, 5701873713553516375488366, -467924697090124685492944896

Offset: 0

Vaclav Kotesovec, Feb 20 2020

sign

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

Cf. A277423, A332692, A332679, A332694, A332695.

Cf. A277392, A295407.

Table[n! * LaguerreL[n, 3*n], {n, 0, 25}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * 3^k * n^k / k!, {k, 0, n}], {n, 1, 25}]}]
Table[n! * Hypergeometric1F1[-n, 1, 3*n], {n, 0, 25}]

a(n) = n!*pollaguerre(n, 0, 3*n); \\ Michel Marcus, Feb 05 2021

A332694 a(n) = (-1)^n * n! * Laguerre(n, 5*n).

Original entry on oeis.org

1, 4, 62, 1614, 58904, 2764880, 158631120, 10755909010, 841471425920, 74605812325020, 7392555309228800, 809594650092540950, 97103822900059929600, 12659189667284189060200, 1782335176686080469555200, 269524635118213823349788250, 43567606796796836119605248000

Offset: 0

Vaclav Kotesovec, Feb 20 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Cf. A277419, A277423, A332692, A332693, A332679, A332695.

Table[(-1)^n * n! * LaguerreL[n, 5*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^(n-k) * 5^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[(-1)^n * n! * Hypergeometric1F1[-n, 1, 5*n], {n, 0, 20}]

a(n) = (-1)^n*n!*pollaguerre(n, 0, 5*n); \\ Michel Marcus, Feb 05 2021

a(n) ~ exp((3-sqrt(5))*n/2) * ((sqrt(5) + 1)/2)^(2*n+1) * n^n / 5^(1/4). - Vaclav Kotesovec, Feb 20 2020, simplified May 09 2021

A332695 a(n) = (-1)^n * n! * Laguerre(n, 6*n).

Original entry on oeis.org

1, 5, 98, 3234, 149784, 8927880, 650696400, 56061791856, 5574017768832, 628158472212096, 79123082415148800, 11015976349601752320, 1679832851707998600192, 278440504042352431942656, 49846084962712218734045184, 9584526091509128369970432000, 1970059291620925696814892810240

Offset: 0

Vaclav Kotesovec, Feb 20 2020

nonn

For m > 4, (-1)^n * n! * Laguerre(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, Table of n, a(n) for n = 0..310

Cf. A277420, A277423, A332692, A332693, A332679, A332694.

Table[(-1)^n * n! * LaguerreL[n, 6*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^(n-k) * 6^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[(-1)^n * n! * Hypergeometric1F1[-n, 1, 6*n], {n, 0, 20}]

a(n) = (-1)^n*n!*pollaguerre(n, 0, 6*n); \\ Michel Marcus, Feb 05 2021

a(n) ~ sqrt(1/2 + 1/sqrt(3)) * 2^n * exp((2-sqrt(3))*n) * ((1 + sqrt(3))/2)^(2*n) * n^n.

A295381 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x/(1 - x))/(1 - x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, -1, 6, 1, -2, -2, -4, 24, 1, -3, -1, -2, -15, 120, 1, -4, 2, 6, 8, -56, 720, 1, -5, 7, 14, 33, 88, -185, 5040, 1, -6, 14, 16, 24, 102, 592, -204, 40320, 1, -7, 23, 6, -31, -104, -9, 3344, 6209, 362880, 1, -8, 34, -22, -120, -380, -1328, -3762, 14464, 112400, 3628800

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

			E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 4*k + 2)*x^2/2! + (-k^3 + 9*k^2 - 18*k + 6)*x^3/3! + (k^4 - 16*k^3 + 72*k^2 - 96*k + 24)*x^4/4! + ...
Square array begins:
    1,   1,   1,    1,    1,    1, ...
    1,   0,  -1,   -2    -3,   -4, ...
    2,  -1,  -2,   -1,    2,    7, ...
    6,  -4,  -2,    6,   14,   16, ...
   24, -15,   8,   33,   24,  -31, ...
  120, -56,  88,  102, -104, -380, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..2 give A000142, A009940, A295382.
Main diagonal gives A277423.
Cf. A289192.

Table[Function[k, n! SeriesCoefficient[Exp[-k x/(1 - x)]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! LaguerreL[n, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! Hypergeometric1F1[-n, 1, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(-k*x/(1 - x))/(1 - x).

A(n,k) = n!*Laguerre(n,k).

cp's OEIS Frontend

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

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

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A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

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A295382 Expansion of e.g.f. exp(-2*x/(1 - x))/(1 - x).

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

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

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A332693 a(n) = n! * Laguerre(n, 3*n).

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A332694 a(n) = (-1)^n * n! * Laguerre(n, 5*n).

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

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