OFFSET
0,2
COMMENTS
The corresponding denominator table is given in A130562.
LINKS
G. C. Greubel, Rows n=0..100 of triangle, 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].
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.
W. Lang, Rational coefficients and more
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
KEYWORD
AUTHOR
Wolfdieter Lang, Aug 07 2007
STATUS
approved