A124324 -id:A124324 - cp's OEIS frontend

A290035 Number of set partitions of [n] having exactly six blocks of size > 1.

10395, 405405, 8828820, 142101960, 1889157270, 21997025050, 232434862660, 2281515816580, 21144158620585, 187205367167455, 1597460349645160, 13226705948208060, 106823347196076588, 845052099612035700, 6569883153685651800, 50334986592592563576

Offset: 12

Alois P. Heinz, Jul 18 2017

Alois P. Heinz, Table of n, a(n) for n = 12..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (84, -3360, 85204, -1538460, 21061260, -227279184, 1984514004, -14280788214, 85828895124, -435042172944, 1872967672764, -6883607484444, 21668771179044, -58531231913904, 135734401224444, -270012108240369, 459750737925864, -667610836187984, 822369705703584, -852988627596768, 737567996531840, -524515347742464, 301116476275200, -135928473663744, 46399971446784, -11247176540160, 1723509964800, -125411328000).

Column k=6 of A124324.

Cf. A290034.

E.g.f.: (exp(x)-x-1)^6/6!*exp(x).
G.f.: -(46416180096*x^15 -267702314880*x^14 +715470788032*x^13 -1174802003648*x^12 +1324630789300*x^11 -1085757157800*x^10 +667971384675*x^9 -313912715655*x^8 +113562125600*x^7 -31611210400*x^6 +6712800710*x^5 -1067591910*x^4 +123053700*x^3 -9702000*x^2 +467775*x -10395)*x^12 / ((7*x-1) *(6*x-1)^2 *(5*x-1)^3 *(4*x-1)^4 *(3*x-1)^5 *(2*x-1)^6 *(x-1)^7).
a(0) = a(1) = 0, for n>1 a(n) = 7*a(n-1) + (n-1)*A290034(n-2). - Ray Chandler, Jul 21 2017

A290036 Number of set partitions of [n] having exactly seven blocks of size > 1.

Original entry on oeis.org

135135, 6756750, 186486300, 3765521760, 62239847670, 893865232260, 11567184248620, 138167790320560, 1549369653596765, 16513475306458130, 168849390493503720, 1668236066705023200, 16016472213542100300, 150103132298249730600, 1378211903535510443400

Offset: 14

Alois P. Heinz, Jul 18 2017

Alois P. Heinz, Table of n, a(n) for n = 14..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (120, -6930, 256564, -6843837, 140161164, -2293167668, 30793317984, -346027498674, 3301174490432, -27034426023228, 191677191769368, -1184495927428914, 6413285791562760, -30547549870770240, 128399094121475760, -477325107218885805, 1571764443755152680, -4588173158058601250, 11875425392771515860, -27240699344951953809, 55318442559624109580, -99273350219483495580, 157041371328829338576, -218253110396224153888, 265336916554318663296, -280638192440433919872, 256449901319079809536, -200704456428999204096, 133025721255740648448, -73584771640934648832, 33313567375875428352, -12012672014150270976, 3315383509586411520, -657169361790566400, 83234996748288000, -5056584744960000).

Column k=7 of A124324.

Cf. A290035.

E.g.f.: (exp(x)-x-1)^7/7!*exp(x).
G.f.: -(1865750631174144*x^21 -13945050326997504*x^20 +49328717299610112*x^19 -109804126032508544*x^18 +172501534253023360*x^17 -203317256909646880*x^16 +186573768183915112*x^15 -136528527507974140*x^14 +80943939197055550*x^13 -39285221171765415*x^12 +15705856242821360*x^11 -5186986300225730*x^10 +1414798298063150*x^9 -317670047760065*x^8 +58326655226840*x^7 -8663283789160*x^6 +1024105011930*x^5 -94030401465*x^4 +6459332880*x^3 -312161850*x^2 +9459450*x -135135)*x^14 / ((8*x-1) *(7*x-1)^2 *(6*x-1)^3 *(5*x-1)^4 *(4*x-1)^5 *(3*x-1)^6 *(2*x-1)^7 *(x-1)^8).
a(0) = a(1) = 0, for n>1 a(n) = 8*a(n-1) + (n-1)*A290035(n-2). - Ray Chandler, Jul 21 2017

A290037 Number of set partitions of [n] having exactly eight blocks of size > 1.

Original entry on oeis.org

2027025, 126351225, 4307428125, 106546244805, 2141473308975, 37150564425975, 577265949054795, 8235084928545475, 109751266389634870, 1384084804861708950, 16677998006092973550, 193476119789167365150, 2173729827868142994450, 23766456155164279406850

Offset: 16

Alois P. Heinz, Jul 18 2017

Alois P. Heinz, Table of n, a(n) for n = 16..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, order 45.

Column k=8 of A124324.

Cf. A290036.

E.g.f.: (exp(x)-x-1)^8/8!*exp(x).

a(0) = a(1) = 0, for n>1 a(n) = 9*a(n-1) + (n-1)*A290036(n-2). - Ray Chandler, Jul 21 2017

A290038 Number of set partitions of [n] having exactly nine blocks of size > 1.

Original entry on oeis.org

34459425, 2618916300, 108030297375, 3211227869850, 77083218186975, 1588144599241200, 29158562820672285, 489227666491814250, 7636058324659014250, 112346788172994575200, 1573773827894456037850, 21155069633041246602700, 274588861338588612866050

Offset: 18

Alois P. Heinz, Jul 18 2017

Alois P. Heinz, Table of n, a(n) for n = 18..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, order 55.

Column k=9 of A124324.

Cf. A290037.

E.g.f.: (exp(x)-x-1)^9/9!*exp(x).

a(0) = a(1) = 0, for n>1 a(n) = 10*a(n-1) + (n-1)*A290037(n-2). - Ray Chandler, Jul 21 2017

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

Alois P. Heinz, Jun 20 2022

			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;
  ...

Alois P. Heinz, Rows n = 0..250, flattened
Wikipedia, Partition of a set

Column k=0 gives A000085.
Row sums give A000110.
T(3n,n) gives A025035.
Cf. A048993, A124324, A124503, A288785.

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

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 *)

Sum_{k=1..n} k * T(n,k) = A288785(n).

A290039 Number of set partitions of [n] having exactly ten blocks of size > 1.

Original entry on oeis.org

654729075, 59580345825, 2924020048950, 102811233675150, 2903837588727075, 70057683857786625, 1499598592952460000, 29215503851264230500, 527544117129699920250, 8948695357270547228350, 144075089938915244609500, 2219478078319305088785500

Offset: 20

Alois P. Heinz, Jul 18 2017

Alois P. Heinz, Table of n, a(n) for n = 20..966
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, order 66.

Column k=10 of A124324.

Cf. A290038.

E.g.f.: (exp(x)-x-1)^10/10!*exp(x).

a(0) = a(1) = 0, for n>1 a(n) = 11*a(n-1) + (n-1)*A290038(n-2). - Ray Chandler, Jul 21 2017

cp's OEIS Frontend

A290035 Number of set partitions of [n] having exactly six blocks of size > 1.

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A290036 Number of set partitions of [n] having exactly seven blocks of size > 1.

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A290037 Number of set partitions of [n] having exactly eight blocks of size > 1.

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A290038 Number of set partitions of [n] having exactly nine blocks of size > 1.

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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.

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A290039 Number of set partitions of [n] having exactly ten blocks of size > 1.

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