A225474 - OEIS

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A225474

Triangle read by rows, k!*2^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, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The Stirling-Frobenius cycle numbers are defined in A225470.

LINKS

Table of n, a(n) for n=0..40.

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, n) ~ A000165; T(n, n-1) ~ A014479.

T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015

EXAMPLE

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

[0] 1,

[1] 1, 2,

[2] 3, 8, 8,

[3] 15, 46, 72, 48,

[4] 105, 352, 688, 768, 384,

[5] 945, 3378, 7600, 11040, 9600, 3840.

MATHEMATICA

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

PROG

(Sage)

@CachedFunction

def SF_CSO(n, k, m):

if k > n or k < 0 : return 0

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

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

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

CROSSREFS

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

Sequence in context: A173162 A198104 A237643 * A368235 A329432 A100805

Adjacent sequences: A225471 A225472 A225473 * A225475 A225476 A225477

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 19 2013

STATUS

approved