A277373 -id:A277373 - cp's OEIS frontend

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

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.

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.

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

cp's OEIS Frontend

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

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

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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).

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