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

%I A289192 #52 Feb 16 2025 08:33:48
%S A289192 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 A289192 1546,720,1,7,47,286,1473,5752,13327,5040,1,8,62,446,2840,14988,58576,
%U A289192 130922,40320,1,9,79,654,4929,32344,173007,671568,1441729,362880
%N A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A289192 Alois P. Heinz, <a href="/A289192/b289192.txt">Antidiagonals n = 0..140, flattened</a>
%H A289192 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289192 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289192 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289192 A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
%F A289192 E.g.f. of column k: exp(k*x/(1-x))/(1-x).
%F A289192 A(n, k) = (-1)^n*KummerU(-n, 1, -k). - _Peter Luschny_, Feb 12 2020
%F A289192 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 A289192 Square array A(n,k) begins:
%e A289192 :   1,    1,    1,     1,     1,     1, ...
%e A289192 :   1,    2,    3,     4,     5,     6, ...
%e A289192 :   2,    7,   14,    23,    34,    47, ...
%e A289192 :   6,   34,   86,   168,   286,   446, ...
%e A289192 :  24,  209,  648,  1473,  2840,  4929, ...
%e A289192 : 120, 1546, 5752, 14988, 32344, 61870, ...
%p A289192 A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
%p A289192 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A289192 A[n_, k_] := n! * LaguerreL[n, -k];
%t A289192 Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 05 2019 *)
%o A289192 (Python)
%o A289192 from sympy import binomial, factorial as f
%o A289192 def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
%o A289192 for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # _Indranil Ghosh_, Jun 28 2017
%o A289192 (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 A289192 (PARI) T(n, k) = n!*pollaguerre(n, 0, -k); \\ _Michel Marcus_, Feb 05 2021
%Y A289192 Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
%Y A289192 Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
%Y A289192 Main diagonal gives A277373.
%Y A289192 Cf. A253286, A341014.
%K A289192 nonn,tabl
%O A289192 0,5
%A A289192 _Alois P. Heinz_, Jun 28 2017