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A225477 Triangle read by rows, 3^k*s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0. 1
1, 2, 3, 10, 21, 9, 80, 198, 135, 27, 880, 2418, 2079, 702, 81, 12320, 36492, 36360, 16065, 3240, 243, 209440, 657324, 727596, 382185, 103275, 13851, 729, 4188800, 13774800, 16523892, 9826488, 3212055, 586845, 56133, 2187, 96342400, 329386800, 421373916, 275580900, 103356729, 23133600, 3051594, 218700, 6561 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, ... (A007494)) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 15 2015

LINKS

Table of n, a(n) for n=0..44.

Peter Luschny, Generalized Eulerian polynomials.

Peter Luschny, The Stirling-Frobenius numbers.

FORMULA

For a recurrence see the Sage program.

T(n,k) = 3^k * A225470(n,k). - Philippe Deléham, May 14 2015.

EXAMPLE

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

[0]      1,

[1]      2,      3,

[2]     10,     21,      9,

[3]     80,    198,    135,     27,

[4]    880,   2418,   2079,    702,     81,

[5]  12320,  36492,  36360,  16065,   3240,   243,

[6] 209440, 657324, 727596, 382185, 103275, 13851, 729.

MATHEMATICA

s[_][0, 0] = 1; s[m_][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];

T[n_, k_] := 3^k*s[3][n, k];

Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

PROG

(Sage)

@CachedFunction

def SF_CS(n, k, m):

    if k > n or k < 0 : return 0

    if n == 0 and k == 0: return 1

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

for n in (0..8): [SF_CS(n, k, 3) for k in (0..n)]

CROSSREFS

T(n, 0) ~ A008544; T(n, n) ~ A000244.

row sums ~ A034000; alternating row sums ~ A008544.

Cf. A225470, A132393 (m=1), A028338 (m=2), A225478 (m=4).

Sequence in context: A141050 A252865 A252868 * A079161 A069565 A139694

Adjacent sequences:  A225474 A225475 A225476 * A225478 A225479 A225480

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 17 2013

STATUS

approved

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Last modified October 18 08:58 EDT 2021. Contains 348067 sequences. (Running on oeis4.)