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A225476 Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0. 1
1, 1, 1, 1, 4, 2, 1, 13, 18, 6, 1, 40, 116, 96, 24, 1, 121, 660, 1020, 600, 120, 1, 364, 3542, 9120, 9480, 4320, 720, 1, 1093, 18438, 74466, 121800, 94920, 35280, 5040, 1, 3280, 94376, 576576, 1394064, 1653120, 1028160, 322560, 40320, 1, 9841, 478440, 4319160 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).

LINKS

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

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} A_2(n, j)*binomial(j, n-k), where A_2(n, j) are the generalized Eulerian numbers of order m=2.

For a recurrence see the Maple program.

EXAMPLE

[n\k][0,   1,   2,    3,   4,   5 ]

[0]   1,

[1]   1,   1,

[2]   1,   4,   2,

[3]   1,  13,  18,    6,

[4]   1,  40, 116,   96,  24,

[5]   1, 121, 660, 1020, 600, 120.

MAPLE

SF_SSO := 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;

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

seq(print(seq(SF_SSO(n, k, 2), 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]]); SFSSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n - k], {j, 0, n}]/m^k; Table[ SFSSO[n, k, 2], {n, 0, 9}, {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_SSO(n, k, m):

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

for n in (0..6): [SF_SSO(n, k, 2) for k in (0..n)]

CROSSREFS

T(n, 0) ~ A000012; T(n, 1) ~ A003462; T(n, 2) ~ A007798.

T(n, n) ~ A000142; T(n, n-1) ~ A001563.

Alternating row sum ~ A000364 (Euler secant numbers).

Cf. A225468, A131689 (m=1).

Sequence in context: A242861 A109244 A171650 * A143777 A236830 A269736

Adjacent sequences:  A225473 A225474 A225475 * A225477 A225478 A225479

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 19 2013

STATUS

approved

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)