A087912 -id:A087912 - cp's OEIS frontend

A277422 a(n) = n!LaguerreL(n, -8n).

1, 9, 322, 19446, 1649688, 180184120, 24070390992, 3801662863152, 692979602529664, 143184960501077376, 33069665092749868800, 8442378658666161822976, 2360674573114695421197312, 717531421372546588398529536, 235551703250624390582942574592

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

In general, if m > 0 and a(n) = n!*LaguerreL(n, -m*n), then a(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.

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

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373 (m=1), A277391 (m=2), A277392 (m=3), A277418 (m=4), A277419 (m=5), A277420 (m=6), A277421 (m=7).

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*(8)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Table[n!*LaguerreL[n, -8*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 8^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0,n, binomial(n,k)*(8)^k*n^k/k!), ", ")) \\ G. C. Greubel, May 16 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 8^k * n^k / k!.

a(n) ~ sqrt(2 + 5/sqrt(6)) * (5 + 2*sqrt(6))^n * exp((-5 + 2*sqrt(6))*n) * n^n / 2.

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

A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

Original entry on oeis.org

1, 5, 34, 286, 2840, 32344, 414160, 5876336, 91356544, 1542401920, 28075364096, 547643910400, 11389266525184, 251428006132736, 5869482147358720, 144413021660821504, 3733822274973040640, 101181690628832198656, 2867011297057247002624, 84764595415605494743040

Offset: 0

Alois P. Heinz, Jun 26 2017

nonn

Number of marriage patterns between a labeled set X of n women and a labeled set Y of n men (all heterosexual): some couples can be formed where one partner is from X and the other from Y, some members of X and Y marry external (unlabeled) partners, and some do not marry.

			a(1) = 5:
[0 0]  [1 0]  [0 1]  [0 0]  [0 1]
[0 0]  [0 0]  [0 0]  [1 0]  [1 0] .
.
a(2) = 34:
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]
.
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[0 0 1]  [0 1 0]  [0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]
[1 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]
[1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]
[0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [1 0 0]
.
[0 1 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]
[1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 1 0]
[0 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]  [0 0 0]  .

Alois P. Heinz, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=4 of A289192.

Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (2*n+3)*a(n-1)-(n-1)^2*a(n-2))
    end:
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):
seq(a(n), n=0..25);
# third Maple program:
a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):
seq(a(n), n=0..25);

Table[n! LaguerreL[n, -4], {n, 0, 30}] (* Indranil Ghosh, Jul 06 2017 *)

from mpmath import *
mp.dps=150
l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))
print([int(fac(i)*l[i]) for i in range(len(l))]) # Indranil Ghosh, Jul 06 2017
# or #
from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -4))
print([a(n) for n in range(31)]) # Indranil Ghosh, Jul 06 2017

E.g.f.: exp(4*x/(1-x))/(1-x).
a(n) = Sum_{i=0..n} i! * (2^(n-i)*binomial(n,i))^2.
a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.
a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).
a(n) = (2*n+3)*a(n-1)-(n-1)^2*a(n-2) for n>=2, a(n) = 4*n+1 for n<2.
a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).
a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 4^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

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

Vaclav Kotesovec, Oct 12 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..417

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

[(&+[Binomial(n,j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Dec 27 2022

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

Table[Sum[Binomial[n, k]^3 * 2^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

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

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

A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.

Original entry on oeis.org

0, 1, 10, 141, 2584, 58745, 1602576, 51165205, 1874935168, 77644293201, 3588075308800, 183111507687581, 10230243235200000, 621111794820235849, 40722033570202507264, 2867494972696071121125, 215840579093024990396416, 17294837586403146090259745, 1469799445329208661211021312

Offset: 0

Peter Luschny, Oct 11 2016

nonn

Cf. A097662, A239768.

Cf. A002720, A087912, A277382.

a := n -> add(binomial(n,n-k)*n^(n-k)*n!/(n-k)!, k=1..n):
seq(a(n), n=0..18);
# Alternatively:
A277372 := n -> n!*LaguerreL(n,-n) - n^n:
seq(simplify(A277372(n)), n=0..18);

a(n) = sum(k=1, n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!); \\ Michel Marcus, Oct 12 2016

a(n) = n!*LaguerreL(n, -n) - n^n.
a(n) = (-1)^n*KummerU(-n, 1, -n) - n^n.
a(n) = n^n*(hypergeom([-n, -n], [], 1/n) - 1) 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

A160615 Numerator of Laguerre(n, -2).

Original entry on oeis.org

1, 3, 7, 43, 27, 719, 3661, 13991, 66769, 133261, 2363513, 116441047, 513267739, 153434147, 96790969339, 2053217625931, 136839921293, 67725860135459, 837687671342383, 7232743280136193, 2996031500521181, 4142815387557270467

Offset: 0

N. J. A. Sloane, Nov 14 2009

			1, 3, 7, 43/3, 27, 719/15, 3661/45, 13991/105, 66769/315, 133261/405, 2363513/4725, 116441047/155925, ...

G. C. Greubel, Table of n, a(n) for n = 0..494

For denominators see A160616.

Cf. A087912.

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

Numerator[Table[LaguerreL[n, -2], {n, 0, 50}]] (* G. C. Greubel, May 06 2018 *)

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

Numerators of coefficients in expansion of exp(2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Aug 29 2018

A160616 Denominator of Laguerre(n, -2).

Original entry on oeis.org

1, 1, 1, 3, 1, 15, 45, 105, 315, 405, 4725, 155925, 467775, 96525, 42567525, 638512875, 30405375, 10854718875, 97692469875, 618718975875, 189403768125, 194896477400625, 238206805711875, 7044115540336875, 8701554491004375

Offset: 0

N. J. A. Sloane, Nov 14 2009

			1, 3, 7, 43/3, 27, 719/15, 3661/45, 13991/105, 66769/315, 133261/405, 2363513/4725, 116441047/155925, ...

Matthew House, Table of n, a(n) for n = 0..506
Eric Weisstein's World of Mathematics, Laguerre Polynomial.

For numerators see A160615.

Cf. A087912.

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

Denominator[Table[LaguerreL[n, -2], {n, 0, 50}]] (* G. C. Greubel, May 06 2018 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*(-2)^k/k!); \\ Michel Marcus, Aug 10 2015

Denominators of coefficients in expansion of exp(2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Aug 29 2018

A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 8, 4, 6, 36, 36, 8, 24, 192, 288, 128, 16, 120, 1200, 2400, 1600, 400, 32, 720, 8640, 21600, 19200, 7200, 1152, 64, 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128, 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

Offset: 0

Peter Luschny, Jan 19 2020

			Triangle starts:
  [0] 1
  [1] 1,     2
  [2] 2,     8,      4
  [3] 6,     36,     36,      8
  [4] 24,    192,    288,     128,     16
  [5] 120,   1200,   2400,    1600,    400,     32
  [6] 720,   8640,   21600,   19200,   7200,    1152,   64
  [7] 5040,  70560,  211680,  235200,  117600,  28224,  3136,   128
  [8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

T(n, 0) = A000142(n), T(n, n) = A000079(n).
Row sums: A087912, alternating row sums: A295382, antidiagonal sums: A222467, positive half sums: A129683, negative half sums: A331334.
Cf. A021009.

A331333 := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, 2*S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
seq(seq(A331333(n, k), k=0..n), n=0..8);

T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
From Peter Bala, Jan 19 2020: (Start)
T(n,k) = 2^k*(n!/k!)*binomial(n,k).
E.g.f.: exp((2*x*t)/(1 - x))/(1 - x) = 1 + (1 + 2*t)*x + (2 + 8*t + 4*t^2)*x^2/2! + .... Cf. A021009. (End)

A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 24, 34, 14, 4, 1, 120, 209, 86, 23, 5, 1, 720, 1546, 648, 168, 34, 6, 1, 5040, 13327, 5752, 1473, 286, 47, 7, 1, 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1, 362880, 1441729, 671568, 173007, 32344, 4929, 654, 79, 9, 1

Offset: 0

Peter Luschny, May 07 2021

			Triangle starts:
0:     1;
1:     1,      1;
2:     2,      2,     1;
3:     6,      7,     3,     1;
4:    24,     34,    14,     4,    1;
5:   120,    209,    86,    23,    5,   1;
6:   720,   1546,   648,   168,   34,   6,  1;
7:  5040,  13327,  5752,  1473,  286,  47,  7,  1;
8: 40320, 130922, 58576, 14988, 2840, 446, 62,  8,  1;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1       2        3        4         5        6
-----------------------------------------------------------------
0:    1,      1,      1,       1,       1,        1,        1, ...
1:    1,      2,      3,       4,       5,        6,        7, ...
2:    2,      7,     14,      23,      34,       47,       62, ...
3:    6,     34,     86,     168,     286,      446,      654, ...
4:   24,    209,    648,    1473,    2840,     4929,     7944, ...
5:  120,   1546,   5752,   14988,   32344,    61870,   108696, ...
6:  720,  13327,  58576,  173007,  414160,   866695,  1649232, ...
7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...

Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
Columns: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Cf. A021009 (Laguerre polynomials), A344048.

T := proc(n, k) option remember;
if n = k then return 1 elif n = k+1 then return k+1 fi;
(2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
seq(print(seq(T(n ,k), k = 0..n)), n = 0..7);

T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Alternative: *)
TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
for(n=0, 9, for(k=0, n, print(T(n, k))))

# Columns of the array.
def column(k, len):
    R. = PowerSeriesRing(QQ, default_prec=len)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()
for col in (0..6): print(column(col, 20))

T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.

A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).

Original entry on oeis.org

1, 1, 3, 1, 4, 14, 1, 5, 22, 86, 1, 6, 32, 152, 648, 1, 7, 44, 248, 1256, 5752, 1, 8, 58, 380, 2248, 12032, 58576, 1, 9, 74, 554, 3768, 23272, 130768, 671568, 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432, 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856

Offset: 0

Detlef Meya, Aug 31 2024

			Triangle starts:
[0] 1;
[1] 1, 3;
[2] 1, 4, 14;
[3] 1, 5, 22, 86;
[4] 1, 6, 32, 152, 648;
[5] 1, 7, 44, 248, 1256, 5752;
[6] 1, 8, 58, 380, 2248, 12032, 58576;
[7] 1, 9, 74, 554, 3768, 23272, 130768, 671568;
[8] 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432;
[9] 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856;
...

Cf. A375855, A000012, A087912 (main diagonal).

T := (n, k) -> 2^k * hypergeom([-n, -k], [], 1/2):
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024

T[n_, k_] := Sum[2^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

from math import isqrt, comb, factorial
def A375854(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    return sum(comb(a,j)*comb(b,j)*factorial(j)<Chai Wah Wu, Nov 13 2024

T(n, k) = Sum_{j=0..k} 2^(k - j)*binomial(n, j)*binomial(k, j)*j!.

cp's OEIS Frontend

A277422 a(n) = n!*LaguerreL(n, -8*n).

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

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A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

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

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A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

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A160615 Numerator of Laguerre(n, -2).

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A160616 Denominator of Laguerre(n, -2).

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A277422 a(n) = n!LaguerreL(n, -8n).

A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)n^(n-k)n!/(n-k)!.