A087912 -id:A087912 - 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

A277382 a(n) = n!*LaguerreL(n, -3).

Original entry on oeis.org

1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164, 8219695239, 148262469336, 2860241078817, 58736954622492, 1278727896354687, 29406849577341552, 712119108949808193, 18108134430393657636, 482306685868464422391, 13425231879291031821576

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

For m > 0, n!*LaguerreL(n, -m) ~ exp(2*sqrt(m*n) - n - m/2) * n^(n + 1/4) / (sqrt(2)*m^(1/4)) * (1 + (3+24*m+4*m^2)/(48*sqrt(m*n))).

Alois P. Heinz, Table of n, a(n) for n = 0..438
W. Van Assche, Erratum to "Weighted zero distribution for polynomials orthogonal on an infinite interval", SIAM J. Math. Anal., 32 (2001), 1169-1170.
Oskar Perron, Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters, Journal für die reine und angewandte Mathematik (1921), vol. 151, p. 63-78.
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A002720, A087912, A277372.
Column k=3 of A289192.
Cf. A160613, A160614.

[Factorial(n)*((&+[Binomial(n,k)*(3^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 09 2018

Table[n!*LaguerreL[n, -3], {n, 0, 20}]
CoefficientList[Series[E^(3*x/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]^2 * 3^k * (n-k)!, {k,0,n}], {n, 0, 20}]

for(n=0,30, print1(n!*(sum(k=0,n, binomial(n,k)*(3^k/k!))), ", ")) \\ G. C. Greubel, May 09 2018

E.g.f.: exp(3*x/(1-x))/(1-x).
a(n) = Sum_{k=0..n} 3^k*(n-k)!*binomial(n, k)^2.
a(n) ~ exp(2*sqrt(3*n)-n-3/2) * n^(n+1/4) / (sqrt(2) * 3^(1/4)) * (1 + 37/(16*sqrt(3*n))).
D-finite with recurrence a(n) = 2*(n+1)*a(n-1) - (n-1)^2*a(n-2).
Lim n -> infinity a(n)/(n!*BesselI(0, 2*sqrt(3*n))) = exp(-3/2).
a(n) = n! * A160613(n)/A160614(n). - Alois P. Heinz, Jun 28 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 3^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jun 28 2017

			Square array A(n,k) begins:
:   1,    1,    1,     1,     1,     1, ...
:   1,    2,    3,     4,     5,     6, ...
:   2,    7,   14,    23,    34,    47, ...
:   6,   34,   86,   168,   286,   446, ...
:  24,  209,  648,  1473,  2840,  4929, ...
: 120, 1546, 5752, 14988, 32344, 61870, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
Main diagonal gives A277373.
Cf. A253286, A341014.

A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := n! * LaguerreL[n, -k];
Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)

{T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

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

from sympy import binomial, factorial as f
def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017

A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
E.g.f. of column k: exp(k*x/(1-x))/(1-x).
A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020
A(n, k) = (2*n+k-1)*A(n-1, k) - (n-1)^2*A(n-2, k) for n > 1. - Seiichi Manyama, Feb 03 2021

A277423 a(n) = n!*LaguerreL(n, n).

Original entry on oeis.org

1, 0, -2, 6, 24, -380, 720, 31794, -361088, -2104056, 101548800, -612792290, -25534891008, 593660731404, 2831189530624, -361541172525750, 4481749181890560, 169464194149739536, -6805365045197340672, -9663483091971306186, 6883830206467440640000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..415
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*(-1)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Table[n!*LaguerreL[n, n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[n! * Hypergeometric1F1[-n, 1, n], {n, 0, 20}] (* Vaclav Kotesovec, Feb 20 2020 *)

a(n) = n!*sum(k=0,n, binomial(n,k)*(-1)^k*n^k/k!); \\ G. C. Greubel, May 16 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * (-1)^k * n^k / k!.
a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Nov 21 2017
a(n) = Sum_{k=0..n} (-n)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 08 2025

A277391 a(n) = n!LaguerreL(n, -2n).

Original entry on oeis.org

1, 3, 34, 654, 17688, 616120, 26252496, 1322624016, 76909665664, 5069558461824, 373529452588800, 30422117430022912, 2713911389090970624, 263171888496899625984, 27563036166079327578112, 3100736138961250867968000, 372888702864658105915244544

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

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

Cf. A002720, A087912, A277382.

Cf. A277373, A277392, A277418, A277419, A277420, A277421, A277422.

[Factorial(n)*(&+[Binomial(n,k)*2^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[n!*LaguerreL[n, -2*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k]*2^k*n^k/k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*2^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

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

a(n) ~ (1 + sqrt(3))^(2*n+1) * n^n / (3^(1/4) * 2^(n+1) * exp((2 - sqrt(3))*n)).

A277392 a(n) = n!LaguerreL(n, -3n).

Original entry on oeis.org

1, 4, 62, 1626, 59928, 2844120, 165100752, 11331597942, 897635712384, 80602042275756, 8090067511468800, 897561658361441106, 109072492644378442752, 14407931244544181001216, 2055559499598438969956352, 314997663481165477898736750, 51601245736595962597616222208

Offset: 0

Vaclav Kotesovec, Oct 12 2016

nonn

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

Cf. A002720, A087912, A277382.

Cf. A277373, A277391, A277418, A277419, A277420, A277421, A277422.

[Factorial(n)*(&+[Binomial(n,k)*3^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[n!*LaguerreL[n, -3*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k]*3^k*n^k/k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*3^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

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

a(n) ~ sqrt(1/2+5/(2*sqrt(21))) * (5+sqrt(21))^n * exp(n*(sqrt(21)-5)/2) * n^n/2^n.

A277418 a(n) = n!LaguerreL(n, -4n).

Original entry on oeis.org

1, 5, 98, 3246, 151064, 9052120, 663449040, 57490690544, 5749754436992, 651830574374784, 82599621627948800, 11569798584488362240, 1775052172071446510592, 296026752508667034942464, 53320241823337034415908864, 10315767337287172256717568000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

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, A277391, A277392, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382, A332679.

[Factorial(n)*(&+[Binomial(n,k)*4^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

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

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*4^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 4^k * n^k / k!.

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

A277419 a(n) = n!LaguerreL(n, -5n).

Original entry on oeis.org

1, 6, 142, 5676, 318744, 23046370, 2038090320, 213094791840, 25714702990720, 3517403388684030, 537798502938028800, 90890936781714193300, 16825134146527678233600, 3385560150770468257273050, 735772370353606135149107200, 171753027520961356975091493000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

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, A277391, A277392, A277418, A277420, A277421, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*5^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

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

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*5^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 5^k * n^k / k!.
a(n) ~ sqrt(1/2 + 7/(6*sqrt(5))) * ((7 + 3*sqrt(5))/2)^n * exp((-7 + 3*sqrt(5))*n/2) * n^n.
Equivalently, a(n) ~ phi^(4*n + 2) * n^n / (sqrt(3) * 5^(1/4) * exp(n/phi^4)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

A277420 a(n) = n!LaguerreL(n, -6n).

Original entry on oeis.org

1, 7, 194, 9078, 596760, 50508120, 5228520912, 639915545808, 90390815432064, 14472947716917120, 2590274418097708800, 512433683486806447872, 111036605823697437490176, 26153418409614396515976192, 6653213794092052464421939200, 1817951594633556391548903168000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

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, A277391, A277392, A277418, A277419, A277421, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*6^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

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

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*6^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 6^k * n^k / k!.

a(n) ~ sqrt(1/2 + 2/sqrt(15)) * (4 + sqrt(15))^n * exp((-4 + sqrt(15))*n) * n^n.

A277421 a(n) = n!LaguerreL(n, -7n).

Original entry on oeis.org

1, 8, 254, 13614, 1025048, 99368620, 11781698256, 1651548277946, 267197019684224, 49000715036948304, 10044513851042988800, 2275926588768085912582, 564838094735322988575744, 152378369304839730672573044, 44397985962782115253758973952

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

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, A277391, A277392, A277418, A277419, A277420, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*7^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[n!*LaguerreL[n, -7*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 7^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*7^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 7^k * n^k / k!.

a(n) ~ sqrt(1/2 + 9/(2*sqrt(77))) * ((9 + sqrt(77))/2)^n * exp((-9 + sqrt(77))*n/2) * n^n.

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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A277382 a(n) = n!*LaguerreL(n, -3).

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A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A277423 a(n) = n!*LaguerreL(n, n).

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A277391 a(n) = n!*LaguerreL(n, -2*n).

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A277392 a(n) = n!*LaguerreL(n, -3*n).

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A277418 a(n) = n!*LaguerreL(n, -4*n).

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

A277391 a(n) = n!LaguerreL(n, -2n).

A277392 a(n) = n!LaguerreL(n, -3n).

A277418 a(n) = n!LaguerreL(n, -4n).

A277419 a(n) = n!LaguerreL(n, -5n).

A277420 a(n) = n!LaguerreL(n, -6n).

A277421 a(n) = n!LaguerreL(n, -7n).