A253286 -id:A253286 - 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

A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).

Original entry on oeis.org

1, 3, 15, 99, 801, 7623, 83079, 1017495, 13808097, 205374123, 3318673599, 57845821707, 1081091446785, 21553820597871, 456410531639799, 10225931132021247, 241609515712343361, 6002109578246918355, 156360266121378584943, 4261404847790207796147

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} m*x^k) = exp(m*x/(1-x)) and m>0, then a(n) ~ n! * m^(1/4) * exp(2*sqrt(m*n) - m/2) / (2 * sqrt(Pi) * n^(3/4)).

G. C. Greubel, Table of n, a(n) for n = 0..435
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A052897, A253286.

[n eq 0 select 1 else 3*Factorial(n-1)*Evaluate(LaguerrePolynomial(n-1, 1), -3): n in [0..25]]; // G. C. Greubel, Feb 24 2021

nmax=20; CoefficientList[Series[Exp[Sum[3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* G. C. Greubel, Feb 24 2021 *)

my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ G. C. Greubel, Feb 05 2017

[1 if n==0 else 3*factorial(n-1)*gen_laguerre(n-1, 1, -3) for n in (0..25)] # G. C. Greubel, Feb 24 2021

E.g.f.: exp(3*x/(1-x)).
a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).
a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Nov 04 2016
From G. C. Greubel, Feb 24 2021: (Start)
a(n) = A253286(n+3, 3).
a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)
For n > 0, a(n) = (n-1)! * Sum_{k=1..n} binomial(n,k) * 3^k / (k-1)!. - Vaclav Kotesovec, Aug 24 2025

A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

Original entry on oeis.org

1, 1, 8, 99, 1696, 37225, 997056, 31535371, 1150303232, 47538819729, 2195314048000, 112032721984051, 6261138045038592, 380309520560089081, 24946892219825709056, 1757549042234670166875, 132356128415391650676736, 10610067001068927596601889, 902057202129607760380428288

Offset: 0

Ilya Gutkovskiy, Oct 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356
Index entries for sequences related to Laguerre polynomials

Main diagonal of A253286.

Cf. A000262, A052857, A052897, A255806, A277372.

[n eq 0 select 1 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 1), -n): n in [0..20]]; // G. C. Greubel, Feb 23 2021

Table[n! SeriesCoefficient[Exp[n x/(1 - x)], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Product[Exp[n x^k], {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[n^k n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 18}]]
Join[{1}, Table[n n! Hypergeometric1F1[1 - n, 2, -n], {n, 1, 18}]]
Table[If[n==0, 1, n!*LaguerreL[n-1, 1, -n]], {n, 0, 20}] (* G. C. Greubel, Feb 23 2021 *)

{a(n) = if(n==0, 1, n!*sum(k=1, n, n^k*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021

a(n) = if (n, n! * pollaguerre(n-1, 1, -n), 1); \\ Michel Marcus, Feb 23 2021

[1 if n==0 else factorial(n)*gen_laguerre(n-1, 1, -n) for n in (0..20)] # G. C. Greubel, Feb 23 2021

a(n) = n! * [x^n] Product_{k>=1} exp(n*x^k).
a(n) ~ exp(n/phi - n) * phi^(2*n) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 01 2017
a(n) = n! * Sum_{k=1..n} n^k * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021
a(n) = n! * LaguerreL(n-1, 1, -n) with a(0) = 1. - G. C. Greubel, Feb 23 2021

A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

Original entry on oeis.org

2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336

Offset: 1

Vladeta Jovovic, Sep 24 2003

Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016

			Triangle begins:
   2;
   4,   4;
  12,  24,  8;
  48, 144, 96, 16;
  ...

G. C. Greubel, Rows n=1..100 of the triangle, flattened
D. Gross and A. Matytsin, Instanton induced large N phase transitions in two and four dimensional QCD, arXiv:hep-th/9404004, 1994.
Index entries for sequences related to Laguerre polynomials

Cf. A008297, A052897 (row sums), A059110, A079621, A105278.

[Factorial(n)*Binomial(n-1,k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> 2*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018

E.g.f.: exp(2*x*y/(1-x)).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)

A256325 a(n) = Sum_{k=0..n-1} (n-k)!exp(-k/2)M_{k-n,1/2}(k), where M is the Whittaker function.

Original entry on oeis.org

0, 0, 1, 5, 24, 136, 933, 7589, 71376, 760796, 9051353, 118784325, 1703388648, 26486926720, 443732646029, 7965563713781, 152504645563072, 3101366761047860, 66753627906345057, 1515914174890163541, 36218232449903567992, 908098606824551207384, 23839591584412453131765

Offset: 0

Peter Luschny, Mar 24 2015

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A253286.

[n eq 0 select 0 else (&+[(n-k-1)*Factorial(k)*Evaluate( LaguerrePolynomial(k, 1), k-n+1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

a := n -> add(exp(-k/2)*WhittakerM(-(n-k),1/2,k)*(n-k)!,k=0..n-1):
seq(round(evalf(a(n),64)), n=0..22);
# Alternatively:
a := n -> add(k*(n-k)!*hypergeom([k-n+1],[2],-k),k=0..n-1):
seq(simplify(a(n)), n=0..22);

Table[Sum[(n-k-1)*k!*LaguerreL[k, 1, k-n+1], {k,0,n-1}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

[sum( (n-k-1)*factorial(k)*gen_laguerre(k, 1, k-n+1) for k in (0..n-1) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

a(n) = Sum_{k=0..n-1} k*(n-k)!*hypergeom([k-n+1],[2],-k).
a(n) = Sum_{k=0..n-1}(Sum_{j=0.. n-k}((n-k-j)!*C(n-k,j)*C(n-k-1,j-1)*k^j)).
a(n) = Sum_{k=0..n-1} (n-k-1)* k! * LaguerreL(k, 1, k-n+1). - G. C. Greubel, Feb 23 2021

A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1

Offset: 0

Seiichi Manyama, Feb 03 2021

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   1,    1,     1,     1,      1, ...
  1,   3,    5,     7,     9,     11, ...
  1,  13,   37,    73,   121,    181, ...
  1,  73,  361,  1009,  2161,   3961, ...
  1, 501, 4361, 17341, 48081, 108101, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.
Main diagonal gives A293146.
Cf. A253286, A341014.

T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)

{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}

{T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}

T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.

T(n,k) = (2*k*n-2*k+1) * T(n-1,k) - k^2 * (n-1) * (n-2) * T(n-2,k) for n > 1.

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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A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).

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A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

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A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

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A256325 a(n) = Sum_{k=0..n-1} (n-k)!*exp(-k/2)*M_{k-n,1/2}(k), where M is the Whittaker function.

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A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

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Mathematica

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A256325 a(n) = Sum_{k=0..n-1} (n-k)!exp(-k/2)M_{k-n,1/2}(k), where M is the Whittaker function.