A277373 -id:A277373 - cp's OEIS frontend

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

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

A295406 a(n) = n! * Laguerre(n, 2*n, -n).

Original entry on oeis.org

1, 4, 58, 1422, 49000, 2174360, 118023264, 7574532826, 561071549056, 47111034709260, 4421715905632000, 458741213603157254, 52129735913348001792, 6439324687323193520608, 859089518697047400878080, 123108032319553206480143250, 18858657171509448248927617024

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295407, A295408.

[Factorial(n)*(&+[Binomial(3*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,2*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[3*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(3*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 2*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(3*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 5/(2*sqrt(13))) * ((11 + sqrt(13))/2)^n * exp((sqrt(13)-5)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(2*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295407 a(n) = n! * Laguerre(n, 3*n, -n).

Original entry on oeis.org

1, 5, 92, 2859, 124832, 7018105, 482598720, 39236322839, 3681751480832, 391611920476653, 46560370087846400, 6119025385880816035, 880818377346674454528, 137824220501484017301281, 23291983597732334528110592, 4228010378355969165140319375

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295408.

[Factorial(n)*(&+[Binomial(4*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,3*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[4*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(4*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

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

a(n) = n!*Sum_{k=0..n} binomial(4*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 3/(2*sqrt(5))) * (8*(sqrt(5)-1))^n * exp((sqrt(5)-3)*n) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(3*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

Original entry on oeis.org

1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373 (m=1), A295385 (m=2), A295406 (m=3), A295407 (m=4).

Cf. A295409, A295418, A332679.

[Factorial(n)*(&+[Binomial(5*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,4*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[5*n,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(5*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 4*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017

A295409 a(n) = n! * Laguerre(n, n^2, -n).

Original entry on oeis.org

1, 3, 58, 2859, 267576, 40818095, 9235507968, 2906955312471, 1215257338052992, 651548571287972859, 435901423022852332800, 356000439852418418920643, 348583395952381998326141952, 403108990190536860168604229031, 543577365164816368801494214352896

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..214
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295407, A295408, A295418.

[Factorial(n)*(&+[Binomial(n*(n+1), n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 11 2018

seq(n!*orthopoly[L](n,n^2,-n),n=0..30); # Robert Israel, Nov 22 2017

Table[n!*LaguerreL[n,n^2,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n*(n+1),n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0,30, binomial(n*(n+1), n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 11 2018

a(n) = n!*pollaguerre(n, n^2, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n! * Sum_{k=0..n} binomial(n*(n+1),n-k)*n^k/k!.
a(n) ~ exp(3/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Nov 23 2017

A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.

Main diagonal of A341014.

[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019

a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

Original entry on oeis.org

1, 2, 32, 1422, 124832, 18246850, 4005713952, 1232956594814, 506672220394496, 267992015325604578, 177340024595660672000, 143531889358151618790862, 139482579412432078779322368, 160267575964062522718064075618, 214924620455826226723051817295872

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..213
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373, A295385, A295406, A295407, A295408, A295409.

[Factorial(n)*(&+[Binomial(n^2, n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, May 13 2018

Table[n!*LaguerreL[n,n*(n-1),-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[n^2,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,25, print1(n!*sum(k=0,n, binomial(n^2, n-k)*n^k/k!), ", ")) \\ G. C. Greubel, May 13 2018

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

a(n) = n! * Sum_{k=0..n} binomial(n^2,n-k)*n^k/k!.
a(n) ~ exp(1/2) * n^(2*n).
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2-n+1). - Ilya Gutkovskiy, Nov 23 2017

A343849 a(n) = Sum_{k = 0..n} n! * LaguerreL(n, -k).

Original entry on oeis.org

1, 3, 23, 294, 5194, 116620, 3175717, 101696700, 3745365444, 155975005998, 7247927859875, 371803988506742, 20870023274690966, 1272424816703533792, 83736949788656865729, 5916106654032037435800, 446636583718649775483144, 35882981062654529341219962, 3056767877633271802374850239

Offset: 0

Peter Luschny, May 08 2021

nonn

Cf. A277373, A343848, A344106, A344107.

a[n_] := Sum[n! LaguerreL[n, -k], {k, 0, n}];
Table[a[n], {n, 0, 18}]

a(n) = n!*sum(m=0, n, sum(j=0, n, binomial(n, j) * m^j / j!))
for(n=0, 18, print(a(n)))

a(n) = (-1)^n * Sum_{k=0..n} KummerU(-n, 1, -k).
a(n) = n! * Sum_{m=0..n} Sum_{j=0..n} binomial(n, j) * m^j / j!.
a(n) ~ exp(n/phi - n) * phi^(2*n+1) * n^n / (5^(1/4) * (1 - exp(-1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 09 2021

A344048 T(n, k) = n! * [x^n] 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, 2, 2, 7, 14, 6, 34, 86, 168, 24, 209, 648, 1473, 2840, 120, 1546, 5752, 14988, 32344, 61870, 720, 13327, 58576, 173007, 414160, 866695, 1649232, 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748

Offset: 0

Peter Luschny, May 08 2021

			Triangle starts:
[0]    1;
[1]    1,      2;
[2]    2,      7,     14;
[3]    6,     34,     86,     168;
[4]   24,    209,    648,    1473,    2840;
[5]  120,   1546,   5752,   14988,   32344,    61870;
[6]  720,  13327,  58576,  173007,  414160,   866695,  1649232;
[7] 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1        2          3           4              5
--------------------------------------------------------------------
[0]   1,      2,      14,       168,        2840,         61870, ...
[1]   1,      7,      86,      1473,       32344,        866695, ...
[2]   2,     34,     648,     14988,      414160,      13373190, ...
[3]   6,    209,    5752,    173007,     5876336,     224995745, ...
[4]  24,   1546,   58576,   2228544,    91356544,    4094022230, ...
[5] 120,  13327,  671568,  31636449,  1542401920,   80031878175, ...
[6] 720, 130922, 8546432, 490102164, 28075364096, 1671426609550, ...

T(n, n) = A277373(n). T(2*n, n) = A344049(n). Row sums are A343849.

Cf. A343847.

# Rows of the array:
A := (n, k) -> (n + k)!*LaguerreL(n + k, -k):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n = 0..6);
# Columns of the array:
egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 16):
C := (k, n) -> (n + k)!*coeff(ser(k), x, n + k):
seq(print(seq(C(k, n), n = 0..6)), k=0..6);

T[n_, k_] := (-1)^(n) HypergeometricU[-n, 1,  -k];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]  // Flatten
(* Alternative: *)
T[n_, k_] := n ! LaguerreL[n , -k];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten

T(n, k) = n! * sum(j=0, n, binomial(n, 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+k)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()[k:]
for col in (0..6): print(column(col, 8))
# Alternative:
@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
A344048 = lambda n, k: factorial(n)*L(n, -k)
print(flatten([[A344048(n, k) for k in (0..n)] for n in (0..7)]))

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

A295384 a(n) = n!Sum_{k=0..n} (-1)^kbinomial(2n,n-k)n^k/k!.

Original entry on oeis.org

1, 1, 0, -15, -112, -135, 9504, 152425, 610560, -27692847, -765107200, -6289891839, 213472972800, 9380264146825, 129378550468608, -3294028613874375, -226623617585053696, -4707649131227927775, 83803818828756418560, 9446689798312021406353, 277055229100887244800000

Offset: 0

Ilya Gutkovskiy, Nov 21 2017

sign

G. C. Greubel, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A006902, A277373, A277423, A295385.

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

a := n -> pochhammer(n, n)*hypergeom([1 - n], [n], n):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023

Table[n! SeriesCoefficient[Exp[-n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 20}]
Table[n! LaguerreL[n, n, n], {n, 0, 20}]
Table[(-1)^n HypergeometricU[-n, n + 1, n], {n, 0, 20}]
Join[{1}, Table[n! Sum[(-1)^k Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 20}]]

for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*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) = Pochhammer(n, n)*hypergeom([1 - n], [n], n). - Peter Luschny, Mar 23 2023

cp's OEIS Frontend

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

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A295406 a(n) = n! * Laguerre(n, 2*n, -n).

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A295407 a(n) = n! * Laguerre(n, 3*n, -n).

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A295408 a(n) = n! * Laguerre(n, 4*n, -n).

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A295409 a(n) = n! * Laguerre(n, n^2, -n).

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A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

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A295418 a(n) = n! * Laguerre(n, n*(n-1), -n).

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

A295384 a(n) = n!Sum_{k=0..n} (-1)^kbinomial(2n,n-k)n^k/k!.