A095421
Triangle read by rows: T(n,m) = number of m-block proper covers (without empty blocks and without multiple blocks) of a labeled n-set (n>=2, 2<=m<=2^n-2).
Original entry on oeis.org
1, 6, 17, 15, 6, 1, 25, 230, 861, 1918, 2975, 3428, 3003, 2002, 1001, 364, 91, 14, 1, 90, 2125, 20930, 127701, 568820, 2003635, 5820750, 14282125, 30030000, 54620475, 86490950, 119759325, 145422600, 155117515, 145422675, 119759850, 86493225
Offset: 2
1;
6,17,15,6,1;
25,230,861,1918,2975,3428,3003,2002,1001,364,91,14,1;
...
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T[n_, m_] := Sum[(-1)^(n - i)*Binomial[n, i]*Binomial[2^i - 1, m], {i, 1, n}] - Binomial[2^n - 2, m - 1]; Table[T[n, m], {n, 2, 10}, {m, 2, 2^n - 2}] // Flatten (* G. C. Greubel, Oct 07 2017 *)
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for(n=2,6, for(m=2, 2^n -2, print1(sum(j=1,n, (-1)^(n-j)* binomial(n, j)*binomial(2^j -1, m)), ", "))) \\ G. C. Greubel, Oct 07 2017
A095153
Number of 4-block covers of a labeled n-set.
Original entry on oeis.org
35, 1225, 24990, 426650, 6779185, 104394675, 1585021340, 23909487700, 359582866335, 5400330984125, 81051093085690, 1216089331752750, 18243600636165485, 273669834496409575, 4105158293128058040, 61578149829707541800, 923677675484159636635
Offset: 3
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[(-50 + 35*3^n - 10*7^n + 15^n)/24 : n in [3..20]]; // Wesley Ivan Hurt, Aug 26 2014
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A095153:=n->(-50+35*3^n-10*7^n+15^n)/24: seq(A095153(n), n=3..20); # Wesley Ivan Hurt, Aug 26 2014
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nn = 19; Table[Sum[(-1)^i Binomial[n, i] Binomial[2^(n - i) - 1, 4], {i, 0, n}], {n, 3, nn}] (* Geoffrey Critzer, Aug 24 2014 *)
Table[(-50 + 35*3^n - 10*7^n + 15^n)/24, {n, 3, 20}] (* Wesley Ivan Hurt, Aug 26 2014 *)
A095155
Number of 6-block covers of a labeled n-set.
Original entry on oeis.org
7, 4977, 711326, 63602770, 4709047749, 320401872035, 20951777849212, 1344192783541860, 85442420316605891, 5406486257577661333, 341342273242841583258, 21527330224106110255670, 1356927944579525164818433, 85508356311211819638169671, 5387705299223777670172444664
Offset: 3
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[(-1764+1624*3^n-735*7^n+175*15^n-21*31^n+63^n)/720 : n in [3..20]]; // Wesley Ivan Hurt, Aug 26 2014
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A095155:=n->(-1764+1624*3^n-735*7^n+175*15^n-21*31^n+63^n)/720: seq(A095155(n), n=3..20); # Wesley Ivan Hurt, Aug 26 2014
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nn = 19; Table[ Sum[(-1)^i Binomial[n, i] Binomial[2^(n - i) - 1, 6], {i, 0, n}], {n, 3, nn}] (* Geoffrey Critzer, Aug 24 2014 *)
Table[(-1764 + 1624*3^n - 735*7^n + 175*15^n - 21*31^n + 63^n)/720, {n, 3, 20}] (* Wesley Ivan Hurt, Aug 26 2014 *)
A095152
Number of 3-block covers of a labeled n-set.
Original entry on oeis.org
1, 32, 321, 2560, 18881, 135072, 954241, 6705920, 47020161, 329377312, 2306349761, 16146574080, 113032395841, 791245902752, 5538778714881, 38771623191040, 271401878897921, 1899814701967392, 13298707562817601, 93090966886860800, 651636810049438401
Offset: 2
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[(11-6*3^n+7^n)/6 : n in [2..30]]; // Wesley Ivan Hurt, Aug 26 2014
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seq((11-6*3^n+7^n)/6, n=2..50); # Robert Israel, Aug 25 2014
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nn = 19; Table[Sum[(-1)^i Binomial[n, i] Binomial[2^(n - i) - 1, 3], {i, 0, n}], {n, 2, nn}] (* Geoffrey Critzer, Aug 24 2014 *)
Table[(11 - 6*3^n + 7^n)/6, {n, 2, 20}] (* Wesley Ivan Hurt, Aug 26 2014 *)
A095154
Number of 5-block covers of a labeled n-set.
Original entry on oeis.org
21, 2919, 155106, 6054006, 208493607, 6791135085, 215553311652, 6758354401932, 210657488261913, 6547648042583571, 203236346721890118, 6304217491485837378, 195489116558570607339, 6061038320388658194777, 187905324183802270088904, 5825262097993829801550744
Offset: 3
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[(274-225*3^n+85*7^n-15*15^n+31^n)/120 : n in [3..20]]; // Wesley Ivan Hurt, Aug 25 2014
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A095154:=n->(274-225*3^n+85*7^n-15*15^n+31^n)/120: seq(A095154(n), n=3..20); # Wesley Ivan Hurt, Aug 25 2014
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nn = 19; Table[Sum[(-1)^i Binomial[n, i] Binomial[2^(n - i) - 1, 5], {i, 0, n}], {n, 3, nn}] (* Geoffrey Critzer, Aug 24 2014 *)
Table[(274 - 225*3^n + 85*7^n - 15*15^n + 31^n)/120, {n, 3, 20}] (* Wesley Ivan Hurt, Aug 25 2014 *)