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A256893 Exponential Riordan array [1, 1/(2-e^x)]. 63
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

LINKS

Table of n, a(n) for n=0..54.

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

EXAMPLE

Number triangle starts:

  1;

  0,    1;

  0,    3,     1;

  0,   13,     9,     1;

  0,   75,    79,    18,    1;

  0,  541,   765,   265,   30,   1;

  ...

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[#])&, 10, True] // Flatten (* Jean-François Alcover, Jul 16 2019, after Sage program *)

PROG

(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)), 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)

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.

Sequence in context: A271704 A307419 A256892 * A137431 A131222 A228334

Adjacent sequences:  A256890 A256891 A256892 * A256894 A256895 A256896

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Apr 17 2015

STATUS

approved

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Last modified October 21 16:50 EDT 2019. Contains 328302 sequences. (Running on oeis4.)