A295381 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x/(1 - x))/(1 - x).

1, 1, 1, 1, 0, 2, 1, -1, -1, 6, 1, -2, -2, -4, 24, 1, -3, -1, -2, -15, 120, 1, -4, 2, 6, 8, -56, 720, 1, -5, 7, 14, 33, 88, -185, 5040, 1, -6, 14, 16, 24, 102, 592, -204, 40320, 1, -7, 23, 6, -31, -104, -9, 3344, 6209, 362880, 1, -8, 34, -22, -120, -380, -1328, -3762, 14464, 112400, 3628800

Offset

0,6

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

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

A(n,k) = n!*Laguerre(n,k).

Example

E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 4*k + 2)*x^2/2! + (-k^3 + 9*k^2 - 18*k + 6)*x^3/3! + (k^4 - 16*k^3 + 72*k^2 - 96*k + 24)*x^4/4! + ...
Square array begins:
    1,   1,   1,    1,    1,    1, ...
    1,   0,  -1,   -2    -3,   -4, ...
    2,  -1,  -2,   -1,    2,    7, ...
    6,  -4,  -2,    6,   14,   16, ...
   24, -15,   8,   33,   24,  -31, ...
  120, -56,  88,  102, -104, -380, ...

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[-k x/(1 - x)]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! LaguerreL[n, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! Hypergeometric1F1[-n, 1, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..2 give A000142, A009940, A295382.

Main diagonal gives A277423.

Cf. A289192.

Sequence in context: A156233 A251725 A292977 * A351420 A331283 A060185

Adjacent sequences: A295378 A295379 A295380 * A295382 A295383 A295384

Keyword

sign,tabl

Author

Ilya Gutkovskiy, Nov 21 2017

Status

approved