login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

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

EXAMPLE

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, ...

MAPLE

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);

MATHEMATICA

A[n_, k_] := n! * LaguerreL[n, -k];

Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, May 05 2019 *)

(* As a triangle read by ascending antidiagonals: *)

a[n_, k_] := (-1)^n HypergeometricU[-n, 1, -k];

Table[Round[a[n - k, k]], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Feb 12 2020 *)

PROG

(Python)

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

CROSSREFS

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.

Sequence in context: A299500 A330141 A007441 * A111933 A144304 A122941

Adjacent sequences:  A289189 A289190 A289191 * A289193 A289194 A289195

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 28 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 30 05:08 EDT 2020. Contains 334712 sequences. (Running on oeis4.)