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
Examples
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, ...
Links
- Eric Weisstein's World of Mathematics, Laguerre Polynomial
- Wikipedia, Laguerre polynomials
- Index entries for sequences related to Laguerre polynomials
Crossrefs
Programs
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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 *) -
PARI
T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);
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PARI
T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ Michel Marcus, Feb 05 2021
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).