A131440 Triangular table of numerators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).

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

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

Sequence in context: A264772 A263917 A324428 * A269950 A190088 A119301

Adjacent sequences: A131437 A131438 A131439 * A131441 A131442 A131443

Keyword

sign,easy,frac,tabl

Author

Wolfdieter Lang, Aug 07 2007

Status

approved