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A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).

%I A119274 #25 Mar 28 2020 14:02:03
%S A119274 1,2,1,12,6,1,120,60,12,1,1680,840,180,20,1,30240,15120,3360,420,30,1,
%T A119274 665280,332640,75600,10080,840,42,1,17297280,8648640,1995840,277200,
%U A119274 25200,1512,56,1,518918400,259459200,60540480,8648640,831600,55440,2520
%N A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).
%C A119274 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.
%C A119274 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
%C A119274 Dividing each diagonal by its initial element generates A054142. - _Tom Copeland_, Oct 10 2016
%F A119274 Number triangle T(n,k) = C(n,k)(2n-k)!/n!.
%F A119274 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
%e A119274 Triangle begins
%e A119274 1,
%e A119274 2, 1,
%e A119274 12, 6, 1,
%e A119274 120, 60, 12, 1,
%e A119274 1680, 840, 180, 20, 1,
%e A119274 30240, 15120, 3360, 420, 30, 1
%p A119274 # The function BellMatrix is defined in A264428.
%p A119274 # Adds (1,0,0,0, ..) as column 0.
%p A119274 BellMatrix(n -> (2*n)!/n!, 9); # _Peter Luschny_, Jan 27 2016
%t A119274 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A119274 rows = 12;
%t A119274 M = BellMatrix[(2#)!/#!&, rows];
%t A119274 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A119274 (Sage) # uses[bell_transform from A264428]
%o A119274 # Adds a column 1,0,0,0,... at the left side of the triangle.
%o A119274 def A119274_row(n):
%o A119274     multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
%o A119274     mfact = [multifact_4_2(k) for k in (0..n)]
%o A119274     return bell_transform(n, mfact)
%o A119274 [A119274_row(n) for n in (0..9)] # _Peter Luschny_, Dec 31 2015
%Y A119274 Cf. A001497, A054142.
%K A119274 easy,nonn,tabl
%O A119274 0,2
%A A119274 _Paul Barry_, May 12 2006