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A225469 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0. 10
1, 3, 1, 9, 10, 1, 27, 79, 21, 1, 81, 580, 310, 36, 1, 243, 4141, 3990, 850, 55, 1, 729, 29230, 48031, 16740, 1895, 78, 1, 2187, 205339, 557571, 299131, 52745, 3689, 105, 1, 6561, 1439560, 6338620, 5044536, 1301286, 137592, 6524, 136, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The definition of the Stirling-Frobenius subset numbers: S_m(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_m(n, j)) / (m^k*k!) where A_m(n, j) are the generalized Eulerian numbers (see the links for details).

This is the Sheffer triangle (exp(3*x),(1/4)*(exp(4*x -1))). See the P. Bala link where this is called exponential Riordan array S_{(4,0,3)}. - Wolfdieter Lang, Apr 13 2017

LINKS

Vincenzo Librandi, Rows n = 0..50, flattened

P. Bala, A 3 parameter family of generalized Stirling numbers.

Peter Luschny, Generalized Eulerian polynomials.

Peter Luschny, The Stirling-Frobenius numbers.

Shi-Mei Ma, Toufik Mansour, Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.

FORMULA

T(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_4(n, j)) / (4^k*k!) where A_4(n,j) = A225118.

For a recurrence see the Maple program.

T(n, 0) ~ A000244; T(n, 1) ~ A016138; T(n, 2) ~ A018054.

T(n, n) ~ A000012; T(n, n-1) ~ A014105.

From Wolfdieter Lang, Apr 13 2017: (Start)

E.g.f.: exp(3*z)*exp((x/4)*(exp(4*z -1))). Sheffer triangle (see a comment above).

E.g.f. column k: exp(3*x)*(exp(4*x) -1)^k/(4^k*k!), k >= 0 (Sheffer property).

O.g.f. column k: x^m/Product_{j=0..k} (1 - (3+4*j)*x), k >= 0.

(End)

EXAMPLE

[n\k][ 0,     1,     2,     3,    4,   5,  6]

[0]    1,

[1]    3,     1,

[2]    9,    10,     1,

[3]   27,    79,    21,     1,

[4]   81,   580,   310,    36,    1,

[5]  243,  4141,  3990,   850,   55,  1,

[6]  729, 29230, 48031, 16740, 1895, 78,  1.

MAPLE

SF_S := proc(n, k, m) option remember;

if n = 0 and k = 0 then return(1) fi;

if k > n or k < 0 then return(0) fi;

SF_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:

seq(print(seq(SF_S(n, k, 4), k=0..n)), n = 0..5);

MATHEMATICA

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/(k!*m^k); Table[ SFS[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, May 29 2013, translated from Sage *)

PROG

(Sage)

@CachedFunction

def EulerianNumber(n, k, m) :

    if n == 0: return 1 if k == 0 else 0

    return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + (m*k+1)*EulerianNumber(n-1, k, m)

def SF_S(n, k, m):

    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/(factorial(k)*m^k)

for n in (0..6): [SF_S(n, k, 4) for k in (0..n)]

CROSSREFS

Cf. A048993 (m=1), A039755 (m=2), A225468 (m=3).

Cf. Columns: A000244, A016138, A018054.

Sequence in context: A105729 A104750 A163394 * A095069 A184061 A222057

Adjacent sequences:  A225466 A225467 A225468 * A225470 A225471 A225472

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Luschny, May 16 2013

STATUS

approved

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Last modified June 17 15:07 EDT 2019. Contains 324185 sequences. (Running on oeis4.)