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A225468 Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0. 11
1, 2, 1, 4, 7, 1, 8, 39, 15, 1, 16, 203, 159, 26, 1, 32, 1031, 1475, 445, 40, 1, 64, 5187, 12831, 6370, 1005, 57, 1, 128, 25999, 107835, 82901, 20440, 1974, 77, 1, 256, 130123, 888679, 1019746, 369061, 53998, 3514, 100, 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. For m = 1 this gives the classical Stirling set numbers A048993. (See the links for details.)

From Peter Bala, Jan 27 2015: (Start)

Exponential Riordan array [ exp(2*z), 1/3*(exp(3*z) - 1)].

Triangle equals P * A111577 = P^(-1) * A075498, where P is Pascal's triangle A007318.

Triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*3^n*binomial((x - 2)/3,n) }n>=0. An example is given below.

This triangle is the particular case a = 3, b = 0, c = 2 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)

LINKS

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

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

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

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_3(n, j)) / (3^k*k!) with A_3(n,j) = A225117.

For a recurrence see the Maple program.

T(n, 0) ~ A000079; T(n, 1) ~ A016127; T(n, 2) ~ A016297; T(n, 3) ~ A025999;

T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024212.

From Peter Bala, Jan 27 2015: (Start)

T(n,k) = sum {i = 0..n} (-1)^(n+i)*3^(i-k)*binomial(n,i)*Stirling2(i+1,k+1).

E.g.f.: exp(2*z)*exp(x/3*(exp(3*z) - 1)) = 1 + (2 + x)*z + (4 + 7*x + x^2)*z^2/2! + ....

T(n,k) = 1/(3^k*k!)*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*(3*j + 2)^n.

O.g.f. for n-th diagonal: exp(-2*x/3)*sum {k >= 0} (3*k + 2)^(k + n - 1)*((x/3*exp(-x))^k)/k!.

O.g.f. column k: 1/( (1 - 2*x)*(1 - 5*x)...(1 - (3*k + 2)*x ). (End)

E.g.f. column k: exp(2*x)*(exp(3*x - 1)/3^k, k >= 0. See the Bala link for the S(3,0,2) exponential Riordan aka Sheffer triangle. - Wolfdieter Lang, Apr 10 2017

EXAMPLE

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

[0]    1,

[1]    2,    1,

[2]    4,    7,     1,

[3]    8,   39,    15,    1,

[4]   16,  203,   159,   26,    1,

[5]   32, 1031,  1475,  445,   40,  1,

[6]   64, 5187, 12831, 6370, 1005, 57,  1.

Connection constants: Row 3: [8, 39, 15, 1] so

x^3 = 8 + 39*(x - 2) + 15*(x - 2)*(x - 5) + (x - 2)*(x - 5)*(x - 8). - Peter Bala, Jan 27 2015

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, 3), 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, 3], {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, 3) for k in (0..n)]

CROSSREFS

Cf. A048993 (m=1), A039755 (m=2), A225469 (m=4).

Cf. A075498, A111577. Columns: A000079, A016127, A016297, A025999. A225466, A225472.

Sequence in context: A059579 A261763 A091320 * A048787 A030102 A072010

Adjacent sequences:  A225465 A225466 A225467 * A225469 A225470 A225471

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Luschny, May 16 2013

STATUS

approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)