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A131440
Triangular table of numerators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).
2
1, 3, -1, 15, -5, 1, 35, -35, 7, -1, 315, -105, 63, -3, 1, 693, -1155, 231, -33, 11, -1, 3003, -3003, 3003, -143, 143, -13, 1, 6435, -15015, 9009, -2145, 715, -13, 1, -1, 109395, -36465, 51051, -7293, 12155, -221, 17, -17, 1, 230945, -692835, 138567, -46189, 46189, -4199, 323, -323, 19, -1
OFFSET
0,2
COMMENTS
The corresponding denominator table is given in A130562.
LINKS
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
FORMULA
a(n,m) = numerator(L(1/2,n,m)) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2, n-m)/m!, n>=m>=0, else 0 (taken in lowest terms).
EXAMPLE
Triangle begins:
[1];
[3,-1];
[15,-5,1];
[35,-35,7,-1];
[315,-105,63,-3,1];
[693,-1155,231,-33,11,-1];
...
Rationals:
[1];
[3/2, -1];
[15/8, -5/2, 1/2];
[35/16, -35/8, 7/4, -1/6];
...
MATHEMATICA
T[n_, k_]:= (-1)^k*Binomial[n+1/2, n-k]/k!; Table[Numerator[T[n, k]], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2018 *)
PROG
(Python)
from sympy import binomial, factorial, Integer
def a(n, m): return ((-1)**m * binomial(n + 1/Integer(2), n -m) / factorial(m)).numerator()
for n in range(21): print([a(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jun 29 2017
(PARI) for(n=0, 10, for(k=0, n, print1(numerator((-1)^k*binomial(n+1/2, n-k)/k!), ", "))) \\ G. C. Greubel, May 14 2018
CROSSREFS
Sequence in context: A264772 A263917 A324428 * A269950 A190088 A119301
KEYWORD
sign,easy,frac,tabl
AUTHOR
Wolfdieter Lang, Aug 07 2007
STATUS
approved