Original entry on oeis.org
0, 0, 1, 3, 12, 50, 225, 1092, 5684, 31572, 186300, 1163085, 7654350, 52928460, 383437327, 2902665885, 22907918640, 188082362120, 1603461748491, 14169892736484, 129594593170210, 1224875863061970, 11948280552370932, 120142063487658003, 1243853543811461148
Offset: 0
-
[seq(binomial(n,2)*combinat[bell](n-2),n=0..50)];
-
Join[{0,0},Table[Binomial[n,2]BellB[n-2],{n,2,30}]] (* Harvey P. Dale, May 06 2014 *)
-
from itertools import count, accumulate, islice
def A105479_gen(): # generator of terms
yield from (0,0,1)
blist, b, c = (1,), 1, 1
for n in count(2):
c += n
blist = list(accumulate(blist, initial=(b:=blist[-1])))
yield b*c
A105479_list = list(islice(A105479_gen(),20)) # Chai Wah Wu, Jun 22 2022
A193274
a(n) = binomial(Bell(n), 2) where B(n) = Bell numbers A000110(n).
Original entry on oeis.org
0, 0, 1, 10, 105, 1326, 20503, 384126, 8567730, 223587231, 6725042325, 230228283165, 8877197732406, 382107434701266, 18221275474580181, 956287167902779240, 54916689705422813731, 3433293323775503064306, 232614384749689991763561, 17010440815323680947084096
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..300
- Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in International Workshop on Combinatorial Algorithms, Victoria, 2011. LNCS.
- Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16, October 2012, Pages 236-248. - From _N. J. A. Sloane_, Oct 03 2012
-
[Binomial(Bell(n),2): n in [0..20]]; // Vincenzo Librandi, Feb 17 2018
-
a:= n-> binomial(combinat[bell](n), 2):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 28 2011
-
a[n_] := With[{b = BellB[n]}, b*(b-1)/2]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 18 2014 *)
-
from itertools import accumulate, islice
def A193274_gen(): # generator of terms
yield 0
blist, b = (1,), 1
while True:
blist = list(accumulate(blist, initial=(b:=blist[-1])))
yield b*(b-1)//2
A193274_list = list(islice(A193274_gen(),30)) # Chai Wah Wu, Jun 22 2022
A152525
a(n) is the number of unordered pairs of disjoint set partitions of an n-element set.
Original entry on oeis.org
0, 0, 1, 7, 65, 811, 12762, 244588, 5574956, 148332645, 4538695461, 157768581675, 6167103354744, 268758895112072, 12961171404183498, 687270616305277589, 39843719438374998543, 2512873126513271758171, 171643113190082528007702, 12647168303374365311984284
Offset: 0
From _Gus Wiseman_, Dec 09 2018: (Start)
The a(3) = 7 unordered pairs:
{{1},{2},{3}}| {{1,2,3}}
{{1},{2,3}} |{{1,2},{3}}
{{1},{2,3}} |{{1,3},{2}}
{{1,2},{3}} |{{1,3},{2}}
{{1},{2,3}} | {{1,2,3}}
{{1,2},{3}} | {{1,2,3}}
{{1,3},{2}} | {{1,2,3}}
(End)
- Alois P. Heinz, Table of n, a(n) for n = 0..200
- Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
- Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16, October 2012, Pages 236-248. - From _N. J. A. Sloane_, Oct 03 2012
Cf.
A000110,
A000258,
A001247,
A008277,
A048993,
A059849,
A060639,
A181939,
A193297,
A318393,
A322441,
A322442,
A320768.
-
a:= n-> add(binomial(n,k)*binomial(combinat[bell](k),2)*
add(Stirling2(n-k,j)*(-1)^j, j=0..n-k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, May 27 2018
-
Array[Sum[Binomial[#, k] Sum[(-1)^j*StirlingS2[# - k, j], {j, 0, # - k}] Binomial[BellB@ k, 2], {k, 0, #}] &, 20, 0] (* Michael De Vlieger, May 27 2018 *)
-
a000110(n) = polcoeff( sum( k=0, n, prod( i=1, k, x / (1 - i*x)), x^n * O(x)), n);
a(n) = sum(k=0, n, binomial(n,k) * sum(j=0, n-k, (-1)^j*stirling(n-k,j, 2) * binomial(a000110(k),2))); \\ Michel Marcus, May 27 2018
Showing 1-3 of 3 results.
Comments