A211700 -id:A211700 - cp's OEIS frontend

A211694 Number of partitions of [n] that contain no isolated singletons.

1, 0, 1, 1, 2, 3, 6, 11, 23, 47, 103, 226, 518, 1200, 2867, 6946, 17234, 43393, 111419, 290242, 768901, 2065172, 5630083, 15549403, 43527487, 123343911, 353864422, 1026935904, 3014535166, 8945274505, 26829206798, 81293234754, 248805520401, 768882019073, 2398686176048, 7552071250781

Offset: 0

R. H. Hardin, Apr 19 2012

nonn

Number of nonnegative integer arrays of length n with new values introduced in order 0 upwards and every value appearing only in runs of at least 2.

Column 2 of A211700.

			All solutions for n = 7:
  0    0    0    0    0    0    0    0    0    0    0
  0    0    0    0    0    0    0    0    0    0    0
  0    0    1    0    1    0    0    0    1    1    1
  1    1    1    1    1    0    0    0    1    1    1
  1    1    2    1    1    0    1    0    0    1    1
  2    1    2    0    2    0    1    1    0    1    0
  2    1    2    0    2    0    1    1    0    1    0

Alois P. Heinz, Table of n, a(n) for n = 0..1132
A. O. Munagi, Set partitions with isolated singletons, Am. Math. Monthly 125 (2018), 447-452.

Cf. A160823, A211700.

Cf. A198648, A198655, A303587, A303588, A363181.

f:=proc(n) local j;
add(combinat:-bell(j-1)*binomial(n-j-1, j-1), j=0..floor(n/2));
end;
[seq(f(n), n=0..100)]; # N. J. A. Sloane, May 19 2018

a[n_] := If[n == 0, 1, Sum[BellB[j-1]*Binomial[n-j-1, j-1], {j, 1, Floor[n/2]}]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 17 2024, after Maple code *)

G.f.: 1+x^2/W(0), where W(k) = 1 - x - x^2/(1 - x^2*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2014

Edited by Andrey Zabolotskiy, Feb 07 2025

A211695 Number of nonnegative integer arrays of length n+2*3-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 3.

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 33, 59, 102, 182, 334, 608, 1121, 2109, 3979, 7564, 14581, 28292, 55245, 108972, 216521, 432839, 872163, 1769716, 3612292, 7422748, 15351002, 31928839, 66803627, 140592917, 297492390, 632896835, 1353709480, 2910262905

Offset: 1

R. H. Hardin Apr 19 2012

nonn

Column 3 of A211700

			All solutions for n=5
..0....0....0....0....0....0....0
..0....0....0....0....0....0....0
..0....0....0....0....0....0....0
..1....1....0....0....0....1....0
..1....1....0....0....0....1....1
..1....1....0....0....1....1....1
..1....0....0....1....1....2....1
..1....0....0....1....1....2....1
..1....0....0....1....1....2....1

R. H. Hardin, Table of n, a(n) for n = 1..210

A211696 Number of nonnegative integer arrays of length n+2*4-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 13, 20, 29, 45, 73, 118, 185, 294, 480, 793, 1298, 2127, 3531, 5932, 9975, 16783, 28415, 48527, 83236, 143064, 246839, 428430, 747193, 1307145, 2294023, 4043525, 7158170, 12715501, 22656789, 40512977, 72713331, 130948115, 236535554

Offset: 1

R. H. Hardin, Apr 19 2012

nonn

Column 4 of A211700.

			All solutions for n=5
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....1
..1....0....0....0....1
..1....0....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A211700.

A211697 Number of nonnegative integer arrays of length n+2*5-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 59, 89, 136, 205, 301, 444, 669, 1026, 1580, 2411, 3666, 5611, 8683, 13542, 21123, 32891, 51299, 80449, 126956, 201015, 318548, 505308, 803817, 1284015, 2058497, 3307355, 5321782, 8579735, 13871094, 22493894

Offset: 1

R. H. Hardin Apr 19 2012

nonn

Column 5 of A211700

			All solutions for n=5
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....1....0
..1....0....0....1....0
..1....1....0....1....0
..1....1....1....1....0
..1....1....1....1....0
..1....1....1....1....0
..1....1....1....1....0
..1....1....1....1....0

R. H. Hardin, Table of n, a(n) for n = 1..210

A211698 Number of nonnegative integer arrays of length n+2*6-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 15, 22, 31, 42, 55, 75, 107, 156, 227, 325, 455, 637, 906, 1312, 1920, 2810, 4077, 5883, 8509, 12407, 18252, 26994, 39910, 58859, 86740, 128153, 190263, 283867, 424664, 635605, 951200, 1424659, 2138744, 3221209, 4865705

Offset: 1

R. H. Hardin, Apr 19 2012

nonn

Column 6 of A211700.

			All solutions for n=5
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....1....0....0....0
..0....1....0....0....1
..0....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1
..1....1....1....0....1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A211700.

A211699 Number of nonnegative integer arrays of length n+2*7-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 16, 23, 32, 43, 56, 71, 93, 127, 178, 251, 351, 483, 652, 878, 1196, 1656, 2323, 3277, 4613, 6441, 8938, 12400, 17294, 24310, 34413, 48895, 69427, 98314, 138953, 196494, 278704, 397034, 567889, 814101, 1167482, 1673334, 2397793

Offset: 1

R. H. Hardin Apr 19 2012

nonn

Column 7 of A211700

			All solutions for n=5
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....1....0
..0....0....0....1....1
..0....1....0....1....1
..0....1....1....1....1
..0....1....1....1....1
..0....1....1....1....1
..0....1....1....1....1
..0....1....1....1....1
..0....1....1....1....1
..0....1....1....1....1

R. H. Hardin, Table of n, a(n) for n = 1..210

cp's OEIS Frontend

A211694 Number of partitions of [n] that contain no isolated singletons.

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A211695 Number of nonnegative integer arrays of length n+2*3-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 3.

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A211696 Number of nonnegative integer arrays of length n+2*4-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 4.

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A211697 Number of nonnegative integer arrays of length n+2*5-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 5.

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A211698 Number of nonnegative integer arrays of length n+2*6-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 6.

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A211699 Number of nonnegative integer arrays of length n+2*7-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least 7.

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