OFFSET
0,2
COMMENTS
n-th numerator of Padé approximation is (1/n!)*sum{j=0..n, C(n,j)(2n-j)!x^j}. Reversal of A113025. Row sums are A001517. First column is A001813. Inverse is A119275.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
Dividing each diagonal by its initial element generates A054142. - Tom Copeland, Oct 10 2016
FORMULA
Number triangle T(n,k) = C(n,k)(2n-k)!/n!.
After adding a leading column (1,0,0,0,...), the triangle gives the coefficients of the Sheffer associated sequence (binomial-type polynomials) for the delta (lowering) operator D(1-D) with e.g.f. exp[ x * (1 - sqrt(1-4t)) / 2 ] . See Mathworld on Sheffer sequences. See A134685 for relation to Catalan numbers. - Tom Copeland, Feb 09 2008
EXAMPLE
Triangle begins
1,
2, 1,
12, 6, 1,
120, 60, 12, 1,
1680, 840, 180, 20, 1,
30240, 15120, 3360, 420, 30, 1
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> (2*n)!/n!, 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(2#)!/#!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
PROG
(Sage) # uses[bell_transform from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
def A119274_row(n):
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
mfact = [multifact_4_2(k) for k in (0..n)]
return bell_transform(n, mfact)
[A119274_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 12 2006
STATUS
approved