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
Examples
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, ...
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
Programs
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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
Formula
E.g.f. of column k: exp(-k*x/(1 - x))/(1 - x).
A(n,k) = n!*Laguerre(n,k).