A275281 -id:A275281 - cp's OEIS frontend

A275283 Number of set partitions of [2n] with symmetric block size list of length n.

1, 1, 3, 19, 171, 2066, 31346, 559987, 11954993, 282835456, 7785919355, 229359684137, 7731656573016, 272633076900991, 10876116332074739, 446659746000614675, 20580725671071449149, 964732749192326683508, 50418595763262446272127, 2656265906893413392905767

Offset: 0

Alois P. Heinz, Jul 21 2016

nonn

			a(0) = 1: {}.
a(1) = 1: 12.
a(2) = 3: 12|34, 13|24, 14|23.
a(3) = 19: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 1|2345|6, 1|2346|5, 15|23|46, 1|2356|4, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 1|2456|3, 16|24|35, 15|26|34, 16|25|34.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Bisection (even part) of A305197.

Cf. A275281.

b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2}]*x^2];
T[n_] := T[n] = Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
a[n_] := T[2n][[n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 21 2021,after Alois P. Heinz in A275281 *)

a(n) = A275281(2n,n).

a(n) ~ c * n^(n-1/2) * d^n / (exp(n) * 2^(n-3/2)), where d = 5.99720652866734051428..., c = 0.331364442872654716... if n is even and c = 0.32118925729236323... if n is odd. - Vaclav Kotesovec, Aug 08 2016

A305197 Number of set partitions of [n] with symmetric block size list of length A004525(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 19, 56, 171, 470, 2066, 10299, 31346, 91925, 559987, 3939653, 11954993, 36298007, 282835456, 2571177913, 7785919355, 24158837489, 229359684137, 2557117944391, 7731656573016, 24350208829581, 272633076900991, 3601150175699409, 10876116332074739

Offset: 0

Alois P. Heinz, May 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Bisections give A275283 (even part), A305198 (odd part).

Cf. A004525, A275281.

b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
a:= n-> coeff(b(n, 0), x, (n+sin(n*Pi/2))/2):
seq(a(n), n=0..30);

b[n_, s_] := b[n, s] = Expand[If[n > s, Binomial[n - 1, n - s - 1]*x, 1] + Sum[Binomial[n - 1, j - 1]*b[n - j, s + j]*Binomial[s + j - 1, j - 1], {j, 1, (n - s)/2}]*x^2];
a[n_] := Coefficient[b[n, 0], x, (n + Sin[n*Pi/2])/2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 13 2018, from Maple *)

a(n) = A275281(n,(n+sin(n*Pi/2))/2).

A275289 Number of set partitions of [n] with symmetric block size list of length three.

Original entry on oeis.org

1, 2, 7, 19, 56, 160, 463, 1337, 3874, 11241, 32682, 95172, 277577, 810706, 2370839, 6941473, 20345618, 59692831, 175295996, 515217034, 1515478535, 4460940067, 13140081770, 38729776774, 114221851951, 337050020750, 995097461503, 2939337252651, 8686270661400

Offset: 3

Alois P. Heinz, Jul 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Wikipedia, Partition of a set

Column k=3 of A275281.

G.f.: -(1/2)*(3*x-1+sqrt((1-3*x)*(x+1)*(2*x-1)^2))/((3*x-1)*(x+1)).
a(n) ~ 3^(n-1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 02 2016
Recurrence: (n-3)*n*a(n) = (n^2 - 3*n + 4)*a(n-1) + (n-2)*(5*n - 11)*a(n-2) + 3*(n-3)*(n-2)*a(n-3). - Vaclav Kotesovec, Aug 02 2016
From Mélika Tebni, Jun 20 2025: (Start)
a(n) = Sum_{k=floor(n/2)..n-2} binomial(n-1, k+1)*binomial(k, n-(k+1)).
Inverse binomial transform of A371965. (End)

A305198 Number of set partitions of [2n+1] with symmetric block size list of length A109613(n).

Original entry on oeis.org

1, 1, 7, 56, 470, 10299, 91925, 3939653, 36298007, 2571177913, 24158837489, 2557117944391, 24350208829581, 3601150175699409, 34626777577615921, 6820331445080882282, 66066554102006208712, 16719951521837764142510, 162903256982698962545956

Offset: 0

Alois P. Heinz, May 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Bisection (odd part) of A305197.

Cf. A109613, A275281, A275283.

b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
a:= n-> coeff(b(2*n+1, 0), x, n+irem(n+1, 2)):
seq(a(n), n=0..20);

b[n_, s_] := b[n, s] = Expand[If[n > s, Binomial[n - 1, n - s - 1] x, 1] + Sum[Binomial[n - 1, j - 1] b[n - j, s + j] Binomial[s + j - 1, j - 1], {j, 1, (n - s)/2}] x^2];
a[n_] := Coefficient[b[2n + 1, 0], x, n + Mod[n + 1, 2]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) = A275281(2n+1,A109613(n)).

A275282 Number of set partitions of [n] with symmetric block size list.

Original entry on oeis.org

1, 1, 2, 2, 7, 9, 47, 80, 492, 985, 7197, 16430, 139316, 361737, 3425683, 9939134, 103484333, 329541459, 3747921857, 12980700318, 159811532315, 598410986533, 7902918548186, 31781977111506, 447462660895105, 1920559118957107, 28699615818386524, 130838216971937408

Offset: 0

Alois P. Heinz, Jul 21 2016

nonn

			a(3) = 2: 123, 1|2|3.
a(4) = 7: 1234, 12|34, 13|24, 14|23, 1|23|4, 1|24|3, 1|2|3|4.
a(5) = 9: 12345, 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Wikipedia, Partition of a set

Row sums of A275281.

b:= proc(n, s) option remember; `if`(n>s,
      binomial(n-1, n-s-1), 1) +add(binomial(n-1, j-1)*
      b(n-j, s+j) *binomial(s+j-1, j-1), j=1..(n-s)/2)
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);

b[n_, s_] := b[n, s] = If[n > s, Binomial[n-1, n-s-1], 1] + Sum[Binomial[n - 1, j - 1]*b[n - j, s + j]*Binomial[s + j - 1, j - 1], {j, 1, (n-s)/2}];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2018, from Maple *)

a(n) = Sum_{k=0..n} A275281(n,k).

A275290 Number of set partitions of [n] with symmetric block size list of length five.

Original entry on oeis.org

1, 3, 22, 86, 470, 2066, 10299, 47503, 229792, 1081851, 5189068, 24671320, 118223765, 565269659, 2713032552, 13022313334, 62644482305, 301600159952, 1454293747517, 7019496361595, 33922782653124, 164099194834163, 794642770006896, 3851565096708617

Offset: 5

Alois P. Heinz, Jul 22 2016

nonn

			a(6) = 3: 1|2|34|5|6, 1|2|35|4|6, 1|2|36|4|5.
a(7) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.

Alois P. Heinz, Table of n, a(n) for n = 5..500
Vaclav Kotesovec, Recurrence (of order 7)
Wikipedia, Partition of a set

Column k=5 of A275281.

a(n) ~ 5^n / (96*Pi*n). - Vaclav Kotesovec, Aug 02 2016

A275291 Number of set partitions of [n] with symmetric block size list of length seven.

Original entry on oeis.org

1, 4, 50, 250, 2160, 12256, 91925, 559987, 3939653, 24964844, 170468820, 1104449861, 7440709045, 48835462696, 327230257098, 2164985118201, 14485751773693, 96353703793654, 644984901718337, 4306885857654070, 28868100135841407, 193352040530826181

Offset: 7

Alois P. Heinz, Jul 22 2016

nonn

			a(8) = 4: 1|2|3|45|6|7|8, 1|2|3|46|5|7|8, 1|2|3|47|5|6|8, 1|2|3|48|5|6|7.

Alois P. Heinz, Table of n, a(n) for n = 7..500
Wikipedia, Partition of a set

Column k=7 of A275281.

a(n) ~ 7^(n+1/2) / (5760*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 02 2016

A275292 Number of set partitions of [n] with symmetric block size list of length nine.

Original entry on oeis.org

1, 5, 95, 575, 7175, 49091, 513206, 3800358, 36298007, 282835456, 2571177913, 20702222482, 183079272792, 1505743012484, 13114463722858, 109381638489849, 944974537003694, 7956650229728706, 68468581589476475, 580345660548809882, 4986413617279183229

Offset: 9

Alois P. Heinz, Jul 22 2016

nonn

			a(10) = 5: 1|2|3|4|56|7|8|9|(10), 1|2|3|4|57|6|8|9|(10), 1|2|3|4|58|6|7|9|(10), 1|2|3|4|59|6|7|8|(10), 1|2|3|4|5(10)|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 9..500
Wikipedia, Partition of a set

Column k=9 of A275281.

a(n) ~ 9^n / (71680*Pi^2*n^2). - Vaclav Kotesovec, Aug 02 2016

A275293 Number of set partitions of [2n] with symmetric block size list of length four.

Original entry on oeis.org

1, 13, 171, 2306, 31795, 446349, 6357295, 91615780, 1333116522, 19555739050, 288834920011, 4291094756898, 64074785496631, 961011037139573, 14469795095794935, 218624167641077960, 3313409217150899536, 50356639055387740752, 767231549954564821746

Offset: 2

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.

Alois P. Heinz, Table of n, a(n) for n = 2..834
Wikipedia, Partition of a set

Bisection of column k=4 of A275281.

a:= proc(n) option remember; `if`(n<3, [0$2, 1, 13][n+1],
      ((n-1)*(4320-23328*n+1365*n^6-11072*n^5+35733*n^4
       -58702*n^3+51744*n^2)*a(n-1)-(4*(2*n-5))*(n-1)*(n-2)
       *(2*n-3)*(21*n^3-55*n^2+44*n-12)*a(n-2))/((2*(n-2))*
        (2*n-1)*(21*n^3-118*n^2+217*n-132)*n^2))
    end:
seq(a(n), n=2..30);

a[2] = 1; a[3] = 13; a[n_] := a[n] = ((n-1)*(4320 - 23328*n + 1365*n^6 - 11072*n^5 + 35733*n^4 - 58702*n^3 + 51744*n^2)*a[n-1] - (4*(2*n-5))*(n-1) *(n-2)*(2*n-3)*(21*n^3 - 55*n^2 + 44*n - 12)*a[n-2])/((2*(n-2))*(2*n-1)* (21*n^3 - 118*n^2 + 217*n - 132)*n^2);
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) ~ 2^(4*n-3) / (3*Pi*n). - Vaclav Kotesovec, Aug 02 2016

A275294 Number of set partitions of [2n] with symmetric block size list of length six.

Original entry on oeis.org

1, 34, 1035, 31346, 958708, 29677572, 929442097, 29416786597, 939795181248, 30273240306244, 982317847204631, 32081167538512119, 1053762976292882820, 34790825470327029730, 1153951843926244071285, 38433994194836318210579, 1284926817226752232868308

Offset: 3

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 1: 1|2|3|4|5|6.
a(4) = 34: 12|3|4|5|6|78, 13|2|4|5|6|78, 1|23|4|5|67|8, 1|23|4|5|68|7, 14|2|3|5|6|78, 1|24|3|5|67|8, 1|24|3|5|68|7, 1|2|34|56|7|8, 1|2|34|57|6|8, 1|2|34|58|6|7, 15|2|3|4|6|78, 1|25|3|4|67|8, 1|25|3|4|68|7, 1|2|35|46|7|8, 1|2|35|47|6|8, 1|2|35|48|6|7, 1|2|36|45|7|8, 1|2|37|45|6|8, 1|2|38|45|6|7, 16|2|3|4|5|78, 1|26|3|4|57|8, 1|26|3|4|58|7, 1|2|36|47|5|8, 1|2|36|48|5|7, 1|2|37|46|5|8, 1|2|38|46|5|7, 1|27|3|4|56|8, 1|28|3|4|56|7, 17|2|3|4|5|68, 1|27|3|4|58|6, 1|2|37|48|5|6, 1|2|38|47|5|6, 1|28|3|4|57|6, 18|2|3|4|5|67.

Alois P. Heinz, Table of n, a(n) for n = 3..300
Vaclav Kotesovec, Recurrence (of order 5)
Wikipedia, Partition of a set

Bisection of column k=6 of A275281.

a(n) ~ 2^(2*n - 6) * 3^(2*n - 1/2) / (5*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 02 2016

cp's OEIS Frontend

A275283 Number of set partitions of [2n] with symmetric block size list of length n.

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A305197 Number of set partitions of [n] with symmetric block size list of length A004525(n).

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A275289 Number of set partitions of [n] with symmetric block size list of length three.

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A305198 Number of set partitions of [2n+1] with symmetric block size list of length A109613(n).

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Maple

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A275282 Number of set partitions of [n] with symmetric block size list.

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A275290 Number of set partitions of [n] with symmetric block size list of length five.

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A275291 Number of set partitions of [n] with symmetric block size list of length seven.

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A275292 Number of set partitions of [n] with symmetric block size list of length nine.

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A275293 Number of set partitions of [2n] with symmetric block size list of length four.

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