A178395 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the inverse Euler polynomial E^{-1}(n,x), 0 <= m <= n.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 14, 28, 35, 28, 14, 4, 1, 1, 9, 18, 42, 63, 63, 42, 18, 9, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1, 1, 6, 33, 110, 495, 396, 462, 396, 495, 110, 33, 6, 1

Offset: 0

Author

Paul Curtz, May 27 2010

Keywords

• nonn
• tabl
• frac

Comments

The triangle of fractions A060096(n,m)/A060097(n,m) contains the coefficients of the Euler Polynomial E(n,x) in row n. The matrix inverse of this triangle is

1;

1/2, 1;

1/2, 1, 1;

1/2, 3/2, 3/2, 1;

1/2, 2, 3, 2, 1;

1/2, 5/2, 5, 5, 5/2, 1;

and defines inverse Euler polynomials E^{-1}(n,x) assuming that row n and column m contain the coefficient [x^m] E^{-1}(n,x). The column m=0 is 1 if n=0, otherwise 1/2.

The current triangle T(n,m) shows the numerator of [x^m] E^{-1}(n,x).

Numerators of exponential Riordan array [(1+exp(x))/2,x]. Central coefficients T(2n,n) are A088218. - Paul Barry, Sep 07 2010

Examples

			From _Paul Barry_, Sep 07 2010: (Start)
Triangle begins
  1;
  1,   1;
  1,   1,   1;
  1,   3,   3,   1;
  1,   2,   3,   2,   1;
  1,   5,   5,   5,   5,   1;
  1,   3,  15,  10,  15,   3,   1;
  1,   7,  21,  35,  35,  21,   7,   1;
  1,   4,  14,  28,  35,  28,  14,   4,   1;
  1,   9,  18,  42,  63,  63,  42,  18,   9,   1;
  1,   5,  45,  60, 105, 126, 105,  60,  45,   5,   1; (End)
		

Links

• T.-X. He, L. C. Hsu, P. J.-S. Shiue, The Sheffer group and the Riordan group, Discr. Appl. Math. 155 (2007) 1895-1909.

Crossrefs

Cf. A178474 (denominators).

Cf. A060096, A060097.

Cf. A088218, A026741, A061041, A064038.

Programs

Maple

nm := 15 : eM := Matrix(nm,nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1,m+1] := coeff(euler(n,x),x,m) ; end do: for m from n+1 to nm-1 do eM[n+1,m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%d,", numer(eM[n,m])) ; end do: end do: # R. J. Mathar, Dec 21 2010

Mathematica

(* The function RiordanArray is defined in A256893. *)
rows = 13;
R = RiordanArray[(1 + E^#)/2&, #&, rows, True];
R // Flatten // Numerator (* Jean-François Alcover, Jul 20 2019 *)

PARI

T(n,k)=numerator((binomial(n,k)+binomial(0,n-k))/2);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print());

Formula

T(n,0) = 1.

T(n,m) = T(n,n-m).

T(n,1) = A026741(n).

T(n,2) = A064038(n) (numerators related to A061041).

Number triangle T(n,k) = [k<=n]*numerator((C(n,k) + C(0,n-k))/2). - Paul Barry, Sep 07 2010