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.

Showing 1-10 of 10 results.

A274538 Number of set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 14, 39, 95, 304, 865, 3103, 10038, 39773, 143473, 620382, 2461099, 11504723, 49658054, 249102263, 1159930119, 6205900348, 30959905841, 175763987955, 934068692102, 5602484594053, 31563436487785, 199267671153562, 1185224170637619
Offset: 0

Views

Author

Alois P. Heinz, Jun 27 2016

Keywords

Comments

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Examples

			a(3) = 2: 13|2, 1|2|3.
a(4) = 3: 13|24, 1|24|3, 1|2|3|4.
a(5) = 7: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
a(6) = 14: 135|246, 13|246|5, 13|24|5|6, 15|246|3, 15|24|3|6, 1|246|35, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|26|3|4|5, 1|2|3|46|5, 1|2|3|4|5|6.

Links

Crossrefs

Row sums of A274537.
Column k=2 of A274835.
Cf. A011655.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1, add(
      `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j, 2] == t, b[n - 1, Max[m, j], 1 - t], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 23 2018, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} A274537(n,k).
a(n) mod 2 = A011655(n) for n>=1.

A274868 Number of set partitions of [n] into exactly four blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 2, 7, 14, 35, 70, 155, 310, 651, 1302, 2667, 5334, 10795, 21590, 43435, 86870, 174251, 348502, 698027, 1396054, 2794155, 5588310, 11180715, 22361430, 44731051, 89462102, 178940587, 357881174, 715795115, 1431590230, 2863245995, 5726491990, 11453115051
Offset: 4

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(6) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
a(7) = 14: 137|24|5|6, 13|24|57|6, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46.

Links

Crossrefs

Column k=4 of A274537.

Programs

  • Mathematica
    Drop[CoefficientList[Series[-x^4/((x - 1) (2 x - 1) (x + 1) (2 x^2 - 1)), {x, 0, 36}], x], 4] (* Michael De Vlieger, Jun 15 2021 *)

Formula

G.f.: -x^4/((x-1)*(2*x-1)*(x+1)*(2*x^2-1)).
From Ridouane Oudra, Jul 13 2023: (Start)
a(n) = x/6 + ((-1)^n - 1)*y, where x = 2^n - 3*sqrt(2)^n + 2 and y = (1/2)*sqrt(2)^(n-1) - (1/4)*sqrt(2)^n - 1/6.
a(n) = ((3 - (-1)^n)/12)*(4^floor(n/2) - 3*2^floor(n/2) + 2).
a(2n) = (4^n)/6 - 2^(n-1) + 1/3.
a(2n+1) = 2*a(2n). (End)

A274869 Number of set partitions of [n] into exactly five blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 2, 13, 26, 113, 226, 833, 1666, 5649, 11298, 36561, 73122, 230161, 460322, 1424401, 2848802, 8720657, 17441314, 53021969, 106043938, 320925969, 641851938, 1936736529, 3873473058, 11665150225, 23330300450, 70169841937, 140339683874, 421734846737
Offset: 5

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(6) = 2: 1|26|3|4|5, 1|2|3|46|5.
a(7) = 13: 13|24|5|6|7, 15|24|3|6|7, 1|24|35|6|7, 15|26|3|4|7, 15|2|3|46|7, 1|26|35|4|7, 1|2|35|46|7, 17|26|3|4|5, 1|26|37|4|5, 1|26|3|4|57, 17|2|3|46|5, 1|2|37|46|5, 1|2|3|46|57.

Links

Crossrefs

Column k=5 of A274537.

Formula

G.f.: x^5/((x-1)*(2*x-1)*(x+1)*(6*x^2-1)*(2*x^2-1)).

A274870 Number of set partitions of [n] into exactly six blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 3, 22, 66, 311, 933, 3632, 10896, 38337, 115011, 381594, 1144782, 3664507, 10993521, 34404964, 103214892, 318365333, 955095999, 2918309966, 8754929898, 26585715663, 79757146989, 241208177496, 723624532488, 2182538747689, 6547616243067, 19713018571138
Offset: 6

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(7) = 3: 17|2|3|4|5|6, 1|2|37|4|5|6, 1|2|3|4|57|6.
a(8) = 22: 13|24|5|6|7|8, 15|24|3|6|7|8, 1|24|35|6|7|8, 15|26|3|4|7|8, 15|2|3|46|7|8, 1|26|35|4|7|8, 1|2|35|46|7|8, 17|26|3|4|5|8, 1|26|37|4|5|8, 1|26|3|4|57|8, 17|2|3|46|5|8, 1|2|37|46|5|8, 1|2|3|46|57|8, 17|28|3|4|5|6, 17|2|3|48|5|6, 17|2|3|4|5|68, 1|28|37|4|5|6, 1|2|37|48|5|6, 1|2|37|4|5|68, 1|28|3|4|57|6, 1|2|3|48|57|6, 1|2|3|4|57|68.

Links

Crossrefs

Column k=6 of A274537.

Formula

G.f.: -x^6/((x-1)*(3*x-1)*(2*x+1)*(2*x-1)*(x+1)*(6*x^2-1)*(2*x^2-1)).

A274871 Number of set partitions of [n] into exactly seven blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 3, 34, 102, 719, 2157, 12260, 36780, 185457, 556371, 2607078, 7821234, 34949443, 104848329, 453798280, 1361394840, 5763944693, 17291834079, 72085646282, 216256938846, 891613471047, 2674840413141, 10940569830060, 32821709490180, 133469376708409
Offset: 7

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(8) = 3: 1|28|3|4|5|6|7, 1|2|3|48|5|6|7, 1|2|3|4|5|68|7.
a(9) = 34: 13|24|5|6|7|8|9, 15|24|3|6|7|8|9, 1|24|35|6|7|8|9, 15|26|3|4|7|8|9, 15|2|3|46|7|8|9, 1|26|35|4|7|8|9, 1|2|35|46|7|8|9, 17|26|3|4|5|8|9, 1|26|37|4|5|8|9, 1|26|3|4|57|8|9, 17|2|3|46|5|8|9, 1|2|37|46|5|8|9, 1|2|3|46|57|8|9, 17|28|3|4|5|6|9, 17|2|3|48|5|6|9, 17|2|3|4|5|68|9, 1|28|37|4|5|6|9, 1|2|37|48|5|6|9, 1|2|37|4|5|68|9, 1|28|3|4|57|6|9, 1|2|3|48|57|6|9, 1|2|3|4|57|68|9, 19|28|3|4|5|6|7, 1|28|39|4|5|6|7, 1|28|3|4|59|6|7, 1|28|3|4|5|6|79, 19|2|3|48|5|6|7, 1|2|39|48|5|6|7, 1|2|3|48|59|6|7, 1|2|3|48|5|6|79, 19|2|3|4|5|68|7, 1|2|39|4|5|68|7, 1|2|3|4|59|68|7, 1|2|3|4|5|68|79.

Links

Crossrefs

Column k=7 of A274537.

Formula

G.f.: x^7/((x-1)*(3*x-1)*(2*x+1)*(2*x-1)*(x+1)*(12*x^2-1) *(6*x^2-1)* (2*x^2-1)).

A274872 Number of set partitions of [n] into exactly eight blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 4, 50, 200, 1519, 6076, 36564, 146256, 770481, 3081924, 14934774, 59739096, 273905827, 1095623308, 4836291512, 19345166048, 83144608885, 332578435540, 1402399388442, 5609597553768, 23330003686119, 93320014744476, 384220628807964, 1536882515231856
Offset: 8

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(8) = 1: 1|2|3|4|5|6|7|8.
a(9) = 4: 19|2|3|4|5|6|7|8, 1|2|39|4|5|6|7|8, 1|2|3|4|59|6|7|8, 1|2|3|4|5|6|79|8.

Links

Crossrefs

Column k=8 of A274537.

Formula

G.f.: -x^8/((x-1)*(3*x-1)*(2*x+1)*(2*x-1)*(3*x+1)*(4*x-1)*(x+1)* (12*x^2-1)*(6*x^2-1)*(2*x^2-1)).

A274873 Number of set partitions of [n] into exactly nine blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 4, 70, 280, 2919, 11676, 94944, 379776, 2669361, 10677444, 68321994, 273287976, 1640345707, 6561382828, 37643205652, 150572822608, 836008721925, 3344034887700, 18122573826942, 72490295307768, 385781480224959, 1543125920899836, 8099850233307144
Offset: 9

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

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

Links

Crossrefs

Column k=9 of A274537.

Formula

G.f.: x^9/((x-1)*(3*x-1)*(2*x+1)*(2*x-1)*(3*x+1)*(4*x-1)*(x+1)* (12*x^2-1)*(6*x^2-1)*(20*x^2-1)*(2*x^2-1)).

A274874 Number of set partitions of [n] into exactly ten blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 5, 95, 475, 5294, 26470, 227294, 1136470, 8351711, 41758555, 277114769, 1385573845, 8568214932, 42841074660, 251848578952, 1259242894760, 7132223195725, 35661115978625, 196428153720067, 982140768600335, 5296485323226634, 26482426616133170
Offset: 10

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(10) = 1: 1|2|3|4|5|6|7|8|9|(10).
a(11) = 5: 1(11)|2|3|4|5|6|7|8|9|(10), 1|2|3(11)|4|5|6|7|8|9|(10), 1|2|3|4|5(11)|6|7|8|9|(10), 1|2|3|4|5|6|7(11)|8|9|(10), 1|2|3|4|5|6|7|8|9(11)|(10).

Links

Crossrefs

Column k=10 of A274537.

Formula

G.f.: -x^10/((x-1)*(4*x+1)*(3*x-1)*(5*x-1)*(2*x+1)*(2*x-1)*(3*x+1)*(4*x-1)*(x+1)*(12*x^2-1)*(6*x^2-1)*(20*x^2-1)*(2*x^2-1)).

A274875 Number of set partitions of [2n] into exactly n blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Original entry on oeis.org

1, 0, 1, 3, 35, 226, 3632, 36780, 770481, 10677444, 277114769, 4882093495, 151410615066, 3238949008872, 116936529884088, 2943848265346408, 121318690777898613, 3513774792447267240, 162794429514873454259, 5332375501915984656921, 274393112247007415104839
Offset: 0

Views

Author

Alois P. Heinz, Jul 09 2016

Keywords

Examples

			a(2) = 1: 13|24.
a(3) = 3: 13|246|5, 15|246|3, 1|246|35.
a(4) = 35: 135|246|7|8, 137|246|5|8, 13|246|57|8, 137|248|5|6, 137|24|5|68, 13|248|57|6, 13|24|57|68, 157|246|3|8, 15|246|37|8, 157|248|3|6, 157|24|3|68, 15|248|37|6, 15|24|37|68, 17|246|35|8, 1|246|357|8, 17|248|35|6, 17|24|35|68, 1|248|357|6, 1|24|357|68, 157|268|3|4, 157|26|3|48, 15|268|37|4, 15|26|37|48, 157|28|3|46, 157|2|3|468, 15|28|37|46, 15|2|37|468, 17|268|35|4, 17|26|35|48, 1|268|357|4, 1|26|357|48, 17|28|35|46, 17|2|35|468, 1|28|357|46, 1|2|357|468.

Links

Crossrefs

Cf. A274537.

Formula

a(n) = A274537(2n,n).

A364267 Total number of blocks in all set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

Original entry on oeis.org

0, 1, 2, 5, 9, 24, 55, 169, 454, 1567, 4823, 18422, 63609, 265929, 1014266, 4595861, 19143089, 93286964, 420483103, 2189786125, 10601936382, 58688597511, 303349005967, 1776842374930, 9754696729753, 60223101819493, 349624680839546, 2267363687696309
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2023

Keywords

Comments

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index. Blocks are ordered with increasing least elements.

Examples

			a(4) = 9 = 2 + 3 + 4: 13|24, 1|24|3, 1|2|3|4.
a(5) = 24 = 2 + 3 + 3 + 3 + 4 + 4 + 5: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.

Links

Crossrefs

Cf. A011655, A274537.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, m, add(
      `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=0..n} k * A274537(n,k).
a(n) mod 2 = A011655(n+1) for n>=1.
Showing 1-10 of 10 results.

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