A124504
Number of partitions of an n-set without blocks of size 3.
Original entry on oeis.org
1, 1, 2, 4, 11, 32, 113, 422, 1788, 8015, 39435, 204910, 1144377, 6722107, 41877722, 273328660, 1875326627, 13427171644, 100415636519, 780856389454, 6312398830812, 52891894374481, 459022366424253, 4117482357137214, 38140612800271305, 364280428671552453, 3584042687233836274
Offset: 0
a(3)=4 because if the set is {1,2,3}, then we have 1|2|3, 1|23, 12|3 and 13|2.
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G:=exp(exp(x)-1-x^3/6): Gser:=series(G,x=0,30): seq(n!*coeff(Gser,x,n),n=0..26);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j=3, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 08 2015, revised, Jun 24 2022
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a[n_] := SeriesCoefficient[Exp[Exp[x]-1-x^3/6], {x, 0, n}]*n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 13 2015 *)
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x='x+O('x^66); Vec(serlaplace( exp(exp(x)-1-x^3/6) ) ) \\ Joerg Arndt, Jan 19 2015
A355144
Number T(n,k) of partitions of [n] having exactly k blocks of size at least three; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.
Original entry on oeis.org
1, 1, 2, 4, 1, 10, 5, 26, 26, 76, 117, 10, 232, 540, 105, 764, 2445, 931, 2620, 11338, 6909, 280, 9496, 53033, 48546, 4900, 35696, 253826, 324753, 64295, 140152, 1235115, 2131855, 691075, 15400, 568504, 6142878, 13792779, 6739876, 400400, 2390480, 31126539, 88890880, 61274213, 7217210
Offset: 0
T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
T(6,2) = 10: 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Triangle T(n,k) begins:
1;
1;
2;
4, 1;
10, 5;
26, 26;
76, 117, 10;
232, 540, 105;
764, 2445, 931;
2620, 11338, 6909, 280;
9496, 53033, 48546, 4900;
35696, 253826, 324753, 64295;
140152, 1235115, 2131855, 691075, 15400;
...
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b:= proc(n) option remember; expand(`if`(n=0, 1, add(
`if`(i>2, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..14); # Alois P. Heinz, Jun 20 2022
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b[n_] := b[n] = Expand[If[n == 0, 1, Sum[If[i > 2, x, 1]*
Binomial[n - 1, i - 1]*b[n - i], {i, 1, n}]]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 25 2022, after Alois P. Heinz *)
A124498
Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} containing k blocks of size 2 (0 <= k <= floor(n/2)).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 6, 6, 3, 17, 20, 15, 53, 90, 45, 15, 205, 357, 210, 105, 871, 1484, 1260, 420, 105, 3876, 7380, 6426, 2520, 945, 18820, 39195, 33390, 18900, 4725, 945, 99585, 213180, 202950, 117810, 34650, 10395, 558847, 1242120, 1293435, 734580, 311850
Offset: 0
T(4,1)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34.
Triangle T(n,k) begins:
: 1;
: 1;
: 1, 1;
: 2, 3;
: 6, 6, 3;
: 17, 20, 15;
: 53, 90, 45, 15;
: 205, 357, 210, 105;
: 871, 1484, 1260, 420, 105;
: 3876, 7380, 6426, 2520, 945;
: 18820, 39195, 33390, 18900, 4725, 945;
: 99585, 213180, 202950, 117810, 34650, 10395;
: 558847, 1242120, 1293435, 734580, 311850, 62370, 10395;
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G:=exp(exp(z)-1+(t-1)*z^2/2): Gser:=simplify(series(G,z=0,16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*`if`(i=2, x^j, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15); # Alois P. Heinz, Mar 08 2015
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d=Exp[Exp[x]-x^2/2!-1]; f[list_] := Select[list,#>0&]; Map[f, Transpose[Table[Range[0,12]! CoefficientList[Series[ x^(2k)/(k! 2!^k) *d, {x,0,12}], x], {k,0,5}]]]//Flatten (* Geoffrey Critzer, Nov 30 2011 *)
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