OFFSET
0,4
COMMENTS
Triangle T(n,k), read by rows: given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 14 2005
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
T(n, k) = (-1)^k*n!*binomial(n, k). - Vladeta Jovovic, May 11 2003
Sum_{k>=0} T(n, k)*T(m, k) = (n+m)!. - Philippe Deléham, Feb 14 2005
A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. - Gerald McGarvey, Aug 20 2009
EXAMPLE
Triangle begins:
1;
1, -1;
2, -4, 2;
6, -18, 18, -6;
24, -96, 144, -96, 24;
...
x^3 = 6*LaguerreL(0,x) - 18*LaguerreL(1,x) + 18*LaguerreL(2,x) - 6*LaguerreL(3,x).
MATHEMATICA
row[n_] := Table[ a[n, k], {k, 0, n}] /. SolveAlways[ x^n == Sum[ a[n, k]*LaguerreL[k, x], {k, 0, n}], x] // First; (* or, after Vladeta Jovovic: *) row[n_] := Table[(-1)^k*n!*Binomial[n, k], {k, 0, n}]; Table[ row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Oct 05 2012 *)
PROG
(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*n!*binomial(n, k), ", "))) \\ G. C. Greubel, Feb 06 2018
(Magma) [[(-1)^k*Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, May 11 2003
STATUS
approved