cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A271792 Number of set partitions of [n] having exactly five pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 21, 242, 2161, 17081, 127540, 931343, 6781012, 49778592, 370879817, 2815885116, 21840126520, 173297901709, 1407962621143, 11717253906772, 99896499620107, 872434202618833, 7803398795633086, 71462039680103117, 669812596793753200, 6423126853283399476
Offset: 5

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Examples

			a(6) = 1: 1|2|3|4|5|6.
a(7) = 21: 12|3|4|5|6|7, 13|24|5|6|7, 1|23|4|5|6|7, 14|25|36|7, 1|24|35|6|7, 1|2|34|5|6|7, 15|26|37|4, 1|25|36|47, 1|2|35|46|7, 1|2|3|45|6|7, 16|27|3|4|5, 1|26|37|4|5, 1|2|36|47|5, 1|2|3|46|57, 1|2|3|4|56|7, 17|2|3|4|5|6, 1|27|3|4|5|6, 1|2|37|4|5|6, 1|2|3|47|5|6, 1|2|3|4|57|6, 1|2|3|4|5|67.

Links

Crossrefs

Column k=5 of A185982.

A271793 Number of set partitions of [n] having exactly six pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 28, 416, 4658, 45095, 404275, 3485508, 29543547, 249503219, 2117198929, 18150345636, 157763663783, 1393693891866, 12533041906408, 114848421469811, 1073139463501186, 10228550424931925, 99466689685460697, 986879431951833062, 9989549096908876766
Offset: 6

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Examples

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

Links

Crossrefs

Column k=6 of A185982.

A271794 Number of set partitions of [n] having exactly seven pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 36, 670, 9187, 106830, 1131496, 11364373, 110881203, 1067003839, 10226994410, 98281964211, 951292607495, 9303462546993, 92137177790612, 925486949131652, 9439151253691761, 97826702119417900, 1030788620292359259, 11046243444215127104, 120413943731688353055
Offset: 7

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Links

Crossrefs

Column k=7 of A185982.

A271795 Number of set partitions of [n] having exactly eight pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 45, 1025, 16886, 232146, 2866698, 33168264, 368982178, 4012603359, 43129752228, 461687900917, 4948321226569, 53305007332086, 578747255834560, 6346136609521608, 70385831812126618, 790497553669858497, 8997272765507744260, 103842387079427555549
Offset: 8

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Links

Crossrefs

Column k=8 of A185982.

A271796 Number of set partitions of [n] having exactly nine pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 55, 1505, 29324, 470327, 6695470, 88385973, 1111798442, 13564841773, 162467818939, 1926323411088, 22746380908378, 268674567620291, 3184862187668777, 37981401796219812, 456537084216273054, 5538881542974933954, 67901571491109849536, 841799517686212572527
Offset: 9

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Links

Crossrefs

Column k=9 of A185982.

A271797 Number of set partitions of [n] having exactly ten pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

0, 1, 66, 2136, 48579, 898863, 14610058, 218221409, 3081371441, 41910602023, 556179920807, 7267182345418, 94111399090828, 1213858926603121, 15651109608458234, 202299948430668867, 2627021681962917991, 34330686032977315128, 452091022434364946290
Offset: 10

Views

Author

Alois P. Heinz, Apr 14 2016

Keywords

Links

Crossrefs

Column k=10 of A185982.

A271841 Number of set partitions of [2n] having exactly n pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.

Original entry on oeis.org

1, 1, 6, 61, 891, 17081, 404275, 11364373, 368982178, 13564841773, 556179920807, 25136678260282, 1240530238800284, 66339010440041817, 3819462133549622416, 235473674234358044731, 15472450628591543437233, 1079168872840695090981865, 79613621745613390178188361
Offset: 0

Views

Author

Alois P. Heinz, Apr 15 2016

Keywords

Links

Crossrefs

Cf. A185982.

Programs

  • Maple
    b:= proc(n, i, m, k) option remember; `if`(k>n, 0, `if`(n=0, 1,
       add(`if`(j=i+1 and k=0, 0, b(n-1, j, max(m, j), k-
      `if`(j=i+1, 1, 0))), j=1..m+1)))
    end:
a:= n-> b(2*n, 1, 0, n):
seq(a(n), n=0..18);
  • Mathematica
    b[n_, i_, m_, k_] := b[n, i, m, k] = If[k > n, 0, If[n == 0, 1, Sum[If[j == i + 1 && k == 0, 0, b[n - 1, j, Max[m, j], k - If[j == i + 1, 1, 0]]], {j, 1, m + 1}]]];
a[n_] := b[2*n, 1, 0, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, May 27 2018, translated from Maple *)

Formula

a(n) = A185982(2n,n).

A272065 Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having not exactly one pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.

Original entry on oeis.org

0, 0, 0, 0, 2, 17, 101, 545, 2935, 16351, 95335, 583373, 3745903, 25208633, 177505205, 1305468285, 10009943248, 79880835800, 662319435622, 5696570446421, 50749156111271, 467630493212126, 4451067568592918, 43709810099960739, 442331477265626019
Offset: 0

Views

Author

Alois P. Heinz, Apr 19 2016

Keywords

Examples

			a(4) = 2: 13|24, 13|2|4.
a(5) = 17: 124|35, 124|3|5, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|3|5, 1|24|35, 1|24|3|5.

Links

Crossrefs

Cf. A000110, A185982, A271271, A272064.

Programs

  • Maple
    b:= proc(n, i, m, l) option remember; `if`(n=0,
      `if`({l[], 1}={1}, 1, 0), add(`if`(j combinat[bell](n)-b(n, 0$2, []):
seq(a(n), n=0..18);
  • Mathematica
    b[n_, i_, m_, l_] := b[n, i, m, l] = If[n == 0, If[Union[Append[l, 1]] == {1}, 1, 0], Sum[If[j < m+1 && j == i+1 && l[[j]] == 1, 0, b[n-1, j, Max[m, j], If[j == m+1, Append[l, If[j == i+1, 1, 0]], If[j == i+1, ReplacePart[l, j -> 1], l]]]], {j, 1, m+1}]]; a[n_] := BellB[n]-b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Formula

a(n) = A000110(n) - A272064(n).

A272105 Number of set partitions of [n] such that for each pair of blocks (b,c) with b

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 41, 115, 362, 1208, 4112, 14107, 49187, 178049
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2016

Keywords

Examples

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 8: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234.
a(5) = 17: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 14|2|35.
a(6) = 41: 123456, 12345|6, 12346|5, 1234|56, 12356|4, 1235|46, 1236|45, 123|456, 12456|3, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 125|3|46, 13456|2, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 13|25|46, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 15|23|46, 145|2|36, 146|2|35, 14|26|35, 14|2|356, 15|24|36, 15|2|346.

Links

Crossrefs

Cf. A000110, A185982, A271270, A272064, A272301.

A272301 Number of set partitions of [n] such that for each pair of blocks (b,c) with b

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 66, 204, 664, 2273, 8283, 32463, 136434, 605848
Offset: 0

Views

Author

Alois P. Heinz, Apr 25 2016

Keywords

Examples

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 14|2|3.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 125|3|4, 1345|2, 134|25, 135|24, 13|245, 13|25|4, 145|23, 14|235, 15|234, 1|2345, 15|23|4, 145|2|3, 14|25|3, 14|2|35, 15|2|34.
a(6) = 66: 123456, 12345|6, 12346|5, 1234|56, 12356|4, 1235|46, 1236|45, 123|456, 1236|4|5, 12456|3, 1245|36, 1246|35, 124|356, 124|36|5, 1256|34, 125|346, 126|345, 12|3456, 126|34|5, 1256|3|4, 125|36|4, 125|3|46, 126|3|45, 13456|2, 1345|26, 1346|25, 134|256, 134|26|5, 1356|24, 135|246, 136|245, 13|2456, 136|24|5, 136|25|4, 13|256|4, 13|25|46, 13|26|45, 1456|23, 145|236, 146|235, 14|2356, 14|236|5, 156|234, 15|2346, 16|2345, 1|23456, 16|234|5, 156|23|4, 15|236|4, 15|23|46, 16|23|45, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 15|24|36, 16|24|35, 156|2|34, 15|26|34, 15|2|346, 16|2|345.

Links

Crossrefs

Cf. A000110, A185982, A271270, A272064, A272105.
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.