A051880
a(n) = binomial(n+4,4)*(2*n+1).
Original entry on oeis.org
1, 15, 75, 245, 630, 1386, 2730, 4950, 8415, 13585, 21021, 31395, 45500, 64260, 88740, 120156, 159885, 209475, 270655, 345345, 435666, 543950, 672750, 824850, 1003275, 1211301, 1452465, 1730575, 2049720, 2414280, 2828936, 3298680, 3828825, 4425015, 5093235
Offset: 0
- Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
- Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.
Cf.
A093645 ((10, 1) Pascal, column m=5).
-
Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],2] (* Harvey P. Dale, Nov 13 2013 *)
A280275
Number of set partitions of [n] where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 1, 2, 5, 12, 37, 118, 387, 1312, 4445, 17034, 73339, 342532, 1616721, 7299100, 31195418, 129179184, 578924785, 3057167242, 18723356715, 120613872016, 738703713245, 4080301444740, 20353638923275, 95273007634552, 443132388701107, 2149933834972928
Offset: 0
a(n) = A000110(n) for n<=3.
a(4) = 12: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 37: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 12|3|4|5, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|23|4|5, 145|2|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
-
with(numtheory):
b:= proc(n, i, s) option remember;
`if`(n=0 or i=1, 1, b(n, i-1, select(x->x<=i-1, s))+
`if`(i>n or factorset(i) intersect s<>{}, 0, b(n-i, i-1,
select(x->x<=i-1, s union factorset(i)))*binomial(n, i)))
end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..30);
-
b[n_, i_, s_] := b[n, i, s] = Expand[If[n==0 || i==1, x^n, b[n, i-1, Select[s, # <= i-1&]] + If[i>n || FactorInteger[i][[All, 1]] ~Intersection~ s != {}, 0, x*b[n-i, i-1, Select[s ~Union~ FactorInteger[i][[All, 1]], # <= i-1&]]*Binomial[n, i]]]];
a[n_] := b[n, n, {}] // CoefficientList[#, x]& // Total;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)
A194924
The number of set partitions of {1,2,...,n} into exactly two subsets A,B such that the greatest common divisor of |A| and |B| = 1.
Original entry on oeis.org
1, 3, 4, 15, 6, 63, 64, 171, 130, 1023, 804, 4095, 2380, 7920, 16384, 65535, 40410, 262143, 246640, 582771, 695860, 4194303, 2884776, 13455325, 11576916, 44739243, 65924824, 268435455, 176422980, 1073741823, 1073741824, 2669774811, 3128164186, 11421338075
Offset: 2
-
a:= n-> `if`(n=2, 1, add(`if`(igcd(k, n-k)=1,
binomial(n, k), 0), k=1..iquo(n, 2))):
seq(a(n), n=2..50); # Alois P. Heinz, Nov 02 2011
-
f[list_]:=x^First[list]/First[list]!+x^Last[list]/Last[list]!;
Prepend[Table[a=Total[Map[f,Select[IntegerPartitions[n,2],Apply[GCD,#]==1&]]];Last[Range[0,n]! CoefficientList[Series[a^2/2!,{x,0,n}],x]],{n,3,30}],1]
(* Second program: *)
a[n_] := If[n == 2, 1, Sum[If[GCD[k, n-k] == 1, Binomial[n, k], 0], {k, 1, Quotient[n, 2]}]];
a /@ Range[2, 50] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)
A280881
Number of set partitions of [n] into exactly three blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 6, 10, 75, 21, 476, 540, 4185, 1375, 47850, 10374, 249431, 395955, 2572680, 278392, 30877389, 5000211, 214159070, 291693150, 1465649955, 1224138223, 23284864476, 13175039700, 157019230225, 198060711381, 1510657970346, 1118209769530, 18176060999625
Offset: 3
a(3) = 1: 1|2|3.
a(4) = 6: 12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
a(7) = 21: 12345|6|7, 12346|5|7, 12347|5|6, 12356|4|7, 12357|4|6, 12367|4|5, 12456|3|7, 12457|3|6, 12467|3|5, 12567|3|4, 13456|2|7, 13457|2|6, 13467|2|5, 13567|2|4, 1|23456|7, 1|23457|6, 1|23467|5, 1|23567|4, 14567|2|3, 1|24567|3, 1|2|34567.
A280882
Number of set partitions of [n] into exactly four blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 10, 20, 245, 56, 2100, 2640, 36795, 8140, 542542, 72436, 3311945, 6050240, 91668080, 2505120, 1972220235, 92327460, 5331136090, 6360277000, 320219686633, 29187262016, 4567704520100, 3125812500200, 39433339579725, 19033980625296, 2173716576665550
Offset: 4
A280883
Number of set partitions of [n] into exactly five blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 15, 35, 630, 126, 6930, 9570, 202455, 35035, 3612609, 361725, 25350780, 50620220, 1262865060, 15864468, 33652405845, 942571665, 68217941715, 72924794635, 7281932609490, 364040548850, 111203641740750, 79780936522950, 2097251547484275, 471615091340211
Offset: 5
A280884
Number of set partitions of [n] into exactly six blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 21, 56, 1386, 252, 18942, 28512, 837837, 122122, 17600583, 1445808, 140411908, 301302288, 10781435088, 79318464, 333208994139, 6813685494, 597189699029, 582524800704, 89346032143830, 3151550453480, 1467268659100530, 1095411461933880, 45383787697612455
Offset: 6
A280885
Number of set partitions of [n] into exactly seven blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 28, 84, 2730, 462, 45276, 73788, 2849847, 365365, 69293224, 4913272, 623100660, 1424294340, 67844069880, 333126696, 2369968906305, 38838853515, 4030108023172, 3637504185700, 767066790520030, 21258921662550, 13536000500234940, 10485013256679780
Offset: 7
A280886
Number of set partitions of [n] into exactly eight blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 36, 120, 4950, 792, 97812, 171600, 8384805, 972400, 233084280, 14734512, 2345611788, 5678805120, 343522980720, 1221436128, 13374187844391, 185339038104, 22390031621100, 18903665638000, 5148292859617050, 118997530835040, 97446697480400580, 78169931776006200
Offset: 8
A280887
Number of set partitions of [n] into exactly nine blocks where sizes of distinct blocks are coprime.
Original entry on oeis.org
1, 45, 165, 8415, 1287, 195195, 366795, 22046310, 2358070, 693870606, 39982878, 7765924530, 19829897010, 1474459789770, 4013220090, 63387586762335, 769009559175, 106739428871075, 85029993637875, 28720696235896665, 574973722599705, 581923529546044725
Offset: 9
Showing 1-10 of 12 results.
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