A289192 - OEIS

%I #52 Feb 16 2025 08:33:48

%S 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,

%T 1546,720,1,7,47,286,1473,5752,13327,5040,1,8,62,446,2840,14988,58576,

%U 130922,40320,1,9,79,654,4929,32344,173007,671568,1441729,362880

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

%H Alois P. Heinz, <a href="/A289192/b289192.txt">Antidiagonals n = 0..140, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).

%F E.g.f. of column k: exp(k*x/(1-x))/(1-x).

%F A(n, k) = (-1)^n*KummerU(-n, 1, -k). - _Peter Luschny_, Feb 12 2020

%F 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

%e Square array A(n,k) begins:

%e :   1,    1,    1,     1,     1,     1, ...

%e :   1,    2,    3,     4,     5,     6, ...

%e :   2,    7,   14,    23,    34,    47, ...

%e :   6,   34,   86,   168,   286,   446, ...

%e :  24,  209,  648,  1473,  2840,  4929, ...

%e : 120, 1546, 5752, 14988, 32344, 61870, ...

%p A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

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

%t Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 05 2019 *)

%o (Python)

%o from sympy import binomial, factorial as f

%o def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))

%o for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # _Indranil Ghosh_, Jun 28 2017

%o (PARI) {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

%o (PARI) T(n, k) = n!*pollaguerre(n, 0, -k); \\ _Michel Marcus_, Feb 05 2021

%Y Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.

%Y Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).

%Y Main diagonal gives A277373.

%Y Cf. A253286, A341014.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Jun 28 2017