A256893 Exponential Riordan array [1, 1/(2-e^x)-1].

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 75, 79, 18, 1, 0, 541, 765, 265, 30, 1, 0, 4683, 8311, 3870, 665, 45, 1, 0, 47293, 100989, 59101, 13650, 1400, 63, 1, 0, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 0, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1

Offset: 0

Author

Peter Luschny, Apr 17 2015

Keywords

• nonn
• tabl

Comments

This is also the matrix product of the Stirling set numbers and the unsigned Lah numbers.

This is also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			Number triangle starts:
  1;
  0,    1;
  0,    3,     1;
  0,   13,     9,     1;
  0,   75,    79,    18,    1;
  0,  541,   765,   265,   30,   1;
  ...
		

Crossrefs

Cf. A088729 which is a variant based on an (1,1)-offset of the number triangles.

Cf. A131222 which is the matrix product of the unsigned Lah numbers and the Stirling cycle numbers.

Cf. A000670, A075729.

Programs

Maple

T:= (n, k)-> n!*coeff(series((1/(2-exp(x))-1)^k/k!, x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 17 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n-1, 1/2)/2, 9); # Peter Luschny, Jan 29 2016

Mathematica

T[n_, k_] := n!*SeriesCoefficient[(1/(2 - Exp[x]) - 1)^k/k!, {x, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)
(* The function BellMatrix is defined in A264428. *)
BellMatrix[PolyLog[-#-1, 1/2]/2&, 9] (* Jean-François Alcover, May 23 2016, after Peter Luschny *)
RiordanArray[d_, h_, n_] := RiordanArray[d, h, n, False];
RiordanArray[d_Function|d_Symbol, h_Function|h_Symbol, n_, exp_:(True | False)] := Module[{M, td, th, k, m},
M[, ] = 0;
td = PadRight[CoefficientList[d[x] + O[x]^n, x], n];
th = PadRight[CoefficientList[h[x] + O[x]^n, x], n];
For[k = 0, k <= n - 1, k++, M[k, 0] = td[[k + 1]]];
For[k = 1, k <= n - 1, k++,
  For[m = k, m <= n - 1, m++,
    M[m, k] = Sum[M[j, k - 1]*th[[m - j + 1]], {j, k - 1, m - 1}]]];
If[exp,
  u = 1;
  For[k = 1, k <= n - 1, k++,
    u *= k;
    For[m = 0, m <= k, m++,
      j = If[m == 0, u, j/m];
      M[k, m] *= j]]];
Table[M[m, k], {m, 0, n - 1}, {k, 0, m}]];
RiordanArray[1&, 1/(2 - Exp[#])-1&, 10, True] // Flatten (* Jean-François Alcover, Jul 16 2019, after Sage program *)

Sage

def riordan_array(d, h, n, exp=false):
    def taylor_list(f,n):
        t = SR(f).taylor(x,0,n-1).list()
        return t + [0]*(n-len(t))
    td = taylor_list(d,n)
    th = taylor_list(h,n)
    M = matrix(QQ,n,n)
    for k in (0..n-1): M[k,0] = td[k]
    for k in (1..n-1):
        for m in (k..n-1):
            M[m,k] = add(M[j,k-1]*th[m-j] for j in (k-1..m-1))
    if exp:
        u = 1
        for k in (1..n-1):
            u *= k
            for m in (0..k):
                j = u if m==0 else j/m
                M[k,m] *= j
    return M
riordan_array(1, 1/(2-exp(x)) - 1, 8, exp=true)
# As a matrix product:
def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, stirling_number2)*matrix(ZZ, 8, Lah)

Formula

Row sums are given by A075729.

T(n,1) = A000670(n) for n>=1.

T(n,k) = n!/k! * [x^n] (1/(2-exp(x))-1)^k. - Alois P. Heinz, Apr 17 2015