A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!*LaguerreL(n, -k*n).

1, 1, 1, 1, 2, 2, 1, 3, 14, 6, 1, 4, 34, 168, 24, 1, 5, 62, 654, 2840, 120, 1, 6, 98, 1626, 17688, 61870, 720, 1, 7, 142, 3246, 59928, 616120, 1649232, 5040, 1, 8, 194, 5676, 151064, 2844120, 26252496, 51988748, 40320, 1, 9, 254, 9078, 318744, 9052120, 165100752, 1322624016, 1891712384, 362880

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

T(n,k) = Sum_{j=0..n} (k*n)^j * (n-j)! * binomial(n,j)^2.

T(n,k) = n! * [x^n] exp(k*n*x/(1-x))/(1-x).

T(n,k) = A289192(n,k*n).

Example

Square array begins:
   1,    1,     1,     1,      1, ...
   1,    2,     3,     4,      5, ...
   2,   14,    34,    62,     98, ...
   6,  168,   654,  1626,   3246, ...
  24, 2840, 17688, 59928, 151064, ...

Mathematica

T[n_, k_] := n! * LaguerreL[n, -k*n]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 05 2021 *)

Prog

(PARI) T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);
(PARI) T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ Michel Marcus, Feb 05 2021

Crossrefs

Columns k=0..8 gives A000142, A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.

Main diagonal gives A340863.

Cf. A021009, A289192 (n!*LaguerreL(n, -k)), A341014.

Sequence in context: A061531 A368093 A368116 * A214722 A071430 A092514

Adjacent sequences: A338432 A338433 A338434 * A338436 A338437 A338438

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 05 2021

Status

approved