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A119274 Triangle of coefficients of numerators in Padé approximation to exp(x). 4
1, 2, 1, 12, 6, 1, 120, 60, 12, 1, 1680, 840, 180, 20, 1, 30240, 15120, 3360, 420, 30, 1, 665280, 332640, 75600, 10080, 840, 42, 1, 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1, 518918400, 259459200, 60540480, 8648640, 831600, 55440, 2520 (list; table; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=0..42.

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

Cf. A001497, A054142.

Sequence in context: A276998 A222866 A008285 * A066991 A132875 A050139

Adjacent sequences:  A119271 A119272 A119273 * A119275 A119276 A119277

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 12 2006

STATUS

approved

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Last modified June 1 13:11 EDT 2020. Contains 334762 sequences. (Running on oeis4.)