A225475 - OEIS

A225475

Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760

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OFFSET

0,4

COMMENTS

The Stirling-Frobenius cycle numbers are defined in A225470.

LINKS

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

Peter Luschny, Generalized Eulerian polynomials.

Peter Luschny, The Stirling-Frobenius numbers.

FORMULA

For a recurrence see the Sage program.

T(n, 0) ~ A001147; T(n, 1) ~ A004041.

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

T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015

EXAMPLE

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

[0] 1,

[1] 1, 1,

[2] 3, 4, 2,

[3] 15, 23, 18, 6,

[4] 105, 176, 172, 96, 24,

[5] 945, 1689, 1900, 1380, 600, 120.

MATHEMATICA

SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2013, translated from Sage *)

PROG

(Sage)

@CachedFunction

def SF_CO(n, k, m):

if k > n or k < 0 : return 0

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

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

for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

CROSSREFS

Cf. A028338, A225479 (m=1), A048594.

Sequence in context: A059114 A246322 A166074 * A259334 A210488 A244364

Adjacent sequences: A225472 A225473 A225474 * A225476 A225477 A225478

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 19 2013

STATUS

approved