A277391 -id:A277391 - 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

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

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.

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

Original entry on oeis.org

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.

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

A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!LaguerreL(n, -kn).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 14, 6, 1, 4, 34, 168, 24, 1, 5, 62, 654, 2840, 120, 1, 6, 98, 1626, 17688, 61870, 720, 1, 7, 142, 3246, 59928, 616120, 1649232, 5040, 1, 8, 194, 5676, 151064, 2844120, 26252496, 51988748, 40320, 1, 9, 254, 9078, 318744, 9052120, 165100752, 1322624016, 1891712384, 362880

Offset: 0

Seiichi Manyama, Feb 05 2021

			Square array begins:
   1,    1,     1,     1,      1, ...
   1,    2,     3,     4,      5, ...
   2,   14,    34,    62,     98, ...
   6,  168,   654,  1626,   3246, ...
  24, 2840, 17688, 59928, 151064, ...

Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..8 gives A000142, A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.
Main diagonal gives A340863.
Cf. A021009, A289192 (n!*LaguerreL(n, -k)), A341014.

T[n_, k_] := n! * LaguerreL[n, -k*n]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 05 2021 *)

T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);

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

T(n,k) = Sum_{j=0..n} (k*n)^j * (n-j)! * binomial(n,j)^2.
T(n,k) = n! * [x^n] exp(k*n*x/(1-x))/(1-x).
T(n,k) = A289192(n,k*n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

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

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).

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

A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!LaguerreL(n, -kn).