A025549
a(n) = (2n-1)!!/lcm{1,3,5,...,2n-1}.
Original entry on oeis.org
1, 1, 1, 1, 3, 3, 3, 45, 45, 45, 945, 945, 4725, 42525, 42525, 42525, 1403325, 49116375, 49116375, 1915538625, 1915538625, 1915538625, 86199238125, 86199238125, 603394666875, 30773128010625, 30773128010625, 1692522040584375, 96473756313309375, 96473756313309375
Offset: 1
Cf.
A196274 (run lengths of equal terms).
-
seq(doublefactorial(2*n-1)/lcm(seq((2*k-1), k=1..n)), n=1..27) ; # Johannes W. Meijer, Jun 08 2009
-
L[ {x___} ] := LCM[ x ]; Table[ (2n-1)!!/L[ Range[ 1, 2n-1, 2 ] ], {n, 1, 50} ]
(* Second program: *)
Array[#!!/LCM @@ Range[1, #, 2] &[2 # - 1] &, 30] (* Michael De Vlieger, Feb 19 2019 *)
-
a(n) = (((2*n)!/n!)/2^n)/lcm(vector(n, i, 2*i-1)); \\ Michel Marcus, Dec 02 2014
A024197
a(n) = 3rd elementary symmetric function of the first n+2 odd positive integers.
Original entry on oeis.org
15, 176, 950, 3480, 10045, 24640, 53676, 106800, 197835, 345840, 576290, 922376, 1426425, 2141440, 3132760, 4479840, 6278151, 8641200, 11702670, 15618680, 20570165, 26765376, 34442500, 43872400, 55361475, 69254640, 85938426, 105844200, 129451505
Offset: 1
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
- Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Equals fourth right hand column of
A028338 triangle.
Equals fourth left hand column of
A109692 triangle.
Equals fourth right hand column of
A161198 triangle divided by 2^m.
(End)
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.
Original entry on oeis.org
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
Offset: 0
[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.
-
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 *)
-
@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)]
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