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 1-6 of 6 results.

A233217 T(n,k)=Number of nXk 0..5 arrays with no element x(i,j) adjacent to value 5-x(i,j) horizontally or antidiagonally, top left element zero, and 1 appearing before 2 3 and 4, and 2 appearing before 3 in row major order.

Original entry on oeis.org

1, 2, 3, 6, 23, 11, 23, 376, 452, 48, 99, 7222, 35446, 10313, 236, 452, 147019, 3054973, 3638416, 249062, 1248, 2136, 3054973, 268289572, 1340889772, 380283286, 6147803, 6896, 10313, 63927526, 23644611625, 496475792293, 591021089923, 39892988056
Offset: 1

Views

Author

R. H. Hardin, Dec 06 2013

Keywords

Comments

Table starts
....1.........2.............6.................23.....................99
....3........23...........376...............7222.................147019
...11.......452.........35446............3054973..............268289572
...48.....10313.......3638416.........1340889772...........496475792293
..236....249062.....380283286.......591021089923........919538740854193
.1248...6147803...39892988056....260625046992322....1703198747507336644
.6896.152986472.4187991850726.114934898294104873.3154729081272072714436

Examples

			Some solutions for n=3 k=4
..0..0..0..1....0..0..0..1....0..0..0..1....0..0..0..1....0..0..0..1
..2..1..2..1....0..1..5..1....0..2..2..1....2..0..0..0....0..2..1..3
..2..0..2..2....1..5..1..0....4..0..1..0....0..1..3..5....2..0..0..2

Links

Crossrefs

Column 1 is A233162(n+1)
Row 1 is A233106

Formula

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3)
k=2: a(n) = 35*a(n-1) -259*a(n-2) +225*a(n-3)
k=3: a(n) = 127*a(n-1) -2331*a(n-2) +2205*a(n-3)
k=4: a(n) = 491*a(n-1) -22099*a(n-2) +21609*a(n-3)
k=5: a(n) = 1975*a(n-1) -228357*a(n-2) +1804281*a(n-3) -4170978*a(n-4) +2593080*a(n-5)
k=6: [order 7]
k=7: [order 11]
Empirical for row n:
n=1: a(n) = 9*a(n-1) -23*a(n-2) +15*a(n-3)
n=2: a(n) = 29*a(n-1) -175*a(n-2) +147*a(n-3) for n>4
n=3: a(n) = 111*a(n-1) -2128*a(n-2) +10532*a(n-3) -17559*a(n-4) +9045*a(n-5) for n>7
n=4: [order 9] for n>12
n=5: [order 19] for n>23
n=6: [order 42] for n>47

A233239 T(n,k)=Number of nXk 0..5 arrays with no element x(i,j) adjacent to value 5-x(i,j) horizontally, diagonally or antidiagonally, top left element zero, and 1 appearing before 2 3 and 4, and 2 appearing before 3 in row major order.

Original entry on oeis.org

1, 2, 3, 6, 19, 11, 23, 271, 313, 48, 99, 4504, 18744, 6046, 236, 452, 79201, 1212549, 1409129, 123352, 1248, 2136, 1419889, 79794804, 338046654, 107709266, 2565169, 6896, 10313, 25622596, 5267525102, 81477098771, 94601758339, 8259321811
Offset: 1

Views

Author

R. H. Hardin, Dec 06 2013

Keywords

Comments

Table starts
.......1............2..................6.......................23
.......3...........19................271.....................4504
......11..........313..............18744..................1212549
......48.........6046............1409129................338046654
.....236.......123352..........107709266..............94601758339
....1248......2565169.........8259321811...........26484848685044
....6896.....53692063.......633724470764.........7415057313896849
...39168...1126297996.....48630297616989......2076029517168733114
..226496..23643610702...3731839458899046....581236325493128357679
.1325568.496455294319.286378755661153351.162731637919752077883024

Examples

			Some solutions for n=3 k=4
..0..1..0..2....0..0..1..2....0..1..0..2....0..0..1..0....0..0..1..2
..3..3..0..0....4..0..0..3....1..0..0..0....2..0..2..1....2..0..1..5
..0..3..3..4....3..0..4..2....2..2..0..2....2..1..2..2....1..5..3..5

Links

Crossrefs

Column 1 is A233162(n+1)
Column 2 is A233107
Row 1 is A233106

Formula

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3)
k=2: a(n) = 29*a(n-1) -175*a(n-2) +147*a(n-3)
k=3: a(n) = 93*a(n-1) -1273*a(n-2) +1943*a(n-3) -882*a(n-4) +120*a(n-5)
k=4: a(n) = 311*a(n-1) -8722*a(n-2) +10022*a(n-3) -1645*a(n-4) +35*a(n-5)
Empirical for row n:
n=1: a(n) = 9*a(n-1) -23*a(n-2) +15*a(n-3)
n=2: a(n) = 23*a(n-1) -81*a(n-2) -143*a(n-3) +82*a(n-4) +120*a(n-5) for n>6
n=3: [order 9] for n>10
n=4: [order 21] for n>22

A233256 T(n,k)=Number of nXk 0..5 arrays with no element x(i,j) adjacent to itself or value 5-x(i,j) horizontally or antidiagonally, top left element zero, and 1 appearing before 2 3 and 4, and 2 appearing before 3 in row major order (unlabelled 6-colorings with no clashing color pairs).

Original entry on oeis.org

1, 1, 3, 3, 10, 11, 10, 104, 136, 48, 36, 1184, 4672, 2080, 236, 136, 13952, 166400, 221696, 32896, 1248, 528, 166400, 6049792, 23896064, 10620928, 524800, 6896, 2080, 1992704, 220626944, 2647261184, 3439984640, 509640704, 8390656, 39168, 8256
Offset: 1

Views

Author

R. H. Hardin, Dec 06 2013

Keywords

Comments

Table starts
.......1...........1................3....................10
.......3..........10..............104..................1184
......11.........136.............4672................166400
......48........2080...........221696..............23896064
.....236.......32896.........10620928............3439984640
....1248......524800........509640704..........495341010944
....6896.....8390656......24461443072........71328837140480
...39168...134225920....1174138781696.....10271348253261824
..226496..2147516416...56358577635328...1479074079750225920
.1325568.34359869440.2705211055407104.212986666384520904704

Examples

			Some solutions for n=3 k=4
..0..1..0..2....0..1..0..2....0..1..2..4....0..1..0..2....0..1..2..4
..5..3..5..2....2..1..0..4....3..4..2..4....3..4..0..4....2..0..3..1
..4..3..1..5....5..2..0..3....3..1..5..4....3..1..5..1....4..0..2..0

Links

Crossrefs

Column 1 is A233162(n+1)
Column 2 is A026244(n-1)
Row 1 is A007582(n-2)

Formula

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3)
k=2: a(n) = 20*a(n-1) -64*a(n-2)
k=3: a(n) = 56*a(n-1) -384*a(n-2)
k=4: a(n) = 160*a(n-1) -2304*a(n-2)
k=5: a(n) = 512*a(n-1) -33792*a(n-2) +589824*a(n-3)
k=6: a(n) = 1664*a(n-1) -471040*a(n-2) +44826624*a(n-3) -1358954496*a(n-4)
k=7: [order 5]
Empirical for row n:
n=1: a(n) = 6*a(n-1) -8*a(n-2) for n>3
n=2: a(n) = 16*a(n-1) -48*a(n-2) for n>3
n=3: a(n) = 48*a(n-1) -448*a(n-2) +1024*a(n-3) for n>5
n=4: a(n) = 160*a(n-1) -6144*a(n-2) +86016*a(n-3) -393216*a(n-4) for n>8
n=5: [order 7] for n>11
n=6: [order 10] for n>16
n=7: [order 28] for n>34

A233174 T(n,k)=Number of nXk 0..5 arrays with no element x(i,j) adjacent to itself or value 5-x(i,j) horizontally, diagonally or antidiagonally, top left element zero, and 1 appearing before 2 3 and 4, and 2 appearing before 3 in row major order (unlabelled 6-colorings with no clashing color pairs).

Original entry on oeis.org

1, 1, 3, 3, 8, 11, 10, 80, 80, 48, 36, 800, 2688, 896, 236, 136, 8576, 78336, 96256, 10496, 1248, 528, 92672, 2469888, 7938048, 3497984, 124928, 6896, 2080, 1009664, 76447744, 736362496, 808583168, 127533056, 1495040, 39168, 8256, 11018240
Offset: 1

Views

Author

R. H. Hardin, Dec 05 2013

Keywords

Comments

Table starts
....1.......1..........3............10................36..................136
....3.......8.........80...........800..............8576................92672
...11......80.......2688.........78336...........2469888.............76447744
...48.....896......96256.......7938048.........736362496..........65265467392
..236...10496....3497984.....808583168......221463445504.......56275748519936
.1248..124928..127533056...82428559360....66799223701504....48667983827959808
.6896.1495040.4653056000.8403942375424.20170789919653888.42129429039341895680

Examples

			Some solutions for n=3 k=4
..0..1..0..2....0..1..2..5....0..1..2..4....0..1..5..2....0..1..2..1
..2..4..0..1....0..1..2..0....0..4..3..1....5..3..0..3....0..1..5..1
..5..1..5..4....5..1..3..5....3..4..5..4....5..4..5..3....0..4..2..4

Links

Crossrefs

Column 1 is A233162(n+1)
Column 2 is A233123
Column 3 is A233124
Row 1 is A007582(n-2)

Formula

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3)
k=2: a(n) = 16*a(n-1) -48*a(n-2)
k=3: a(n) = 48*a(n-1) -448*a(n-2) +1024*a(n-3)
k=4: a(n) = 128*a(n-1) -2816*a(n-2) +16384*a(n-3)
k=5: [order 7]
k=6: [order 10]
k=7: [order 20]
Empirical for row n:
n=1: a(n) = 6*a(n-1) -8*a(n-2) for n>3
n=2: a(n) = 12*a(n-1) -128*a(n-3) for n>4
n=3: a(n) = 32*a(n-1) +64*a(n-2) -3072*a(n-3) +8192*a(n-4) for n>5
n=4: [order 7] for n>8
n=5: [order 10] for n>11
n=6: [order 24] for n>25
n=7: [order 47] for n>48

A233202 T(n,k)=Number of nXk 0..7 arrays with no element x(i,j) adjacent to itself or value 7-x(i,j) horizontally, antidiagonally or vertically, top left element zero, and 1 appearing before 2 3 4 5 and 6, 2 appearing before 3 4 and 5, and 3 appearing before 4 in row major order (unlabelled 8-colorings with no clashing color pairs).

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 11, 36, 36, 11, 48, 528, 1440, 528, 48, 236, 8256, 62720, 62720, 8256, 236, 1248, 131328, 2779136, 7802880, 2779136, 131328, 1248, 6896, 2098176, 123551744, 981532672, 981532672, 123551744, 2098176, 6896, 39168, 33558528
Offset: 1

Views

Author

R. H. Hardin, Dec 05 2013

Keywords

Comments

Table starts
....1.......1..........3.............11................48...................236
....1.......3.........36............528..............8256................131328
....3......36.......1440..........62720...........2779136.............123551744
...11.....528......62720........7802880.........981532672..........123833679872
...48....8256....2779136......981532672......352524959744.......127365174788096
..236..131328..123551744...123833679872...127365174788096....132364161848967168
.1248.2098176.5496242176.15635845218304.46111939268444160.138127371171167993856

Examples

			Some solutions for n=3 k=4
..0..1..2..7....0..1..2..0....0..1..2..3....0..1..2..7....0..1..7..2
..3..7..4..5....5..3..6..3....5..4..6..5....5..7..3..6....5..3..6..4
..6..5..7..6....6..7..2..6....6..2..3..6....3..5..0..3....6..7..2..6

Links

Crossrefs

Column 1 is A233162

Formula

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3) for n>4
k=2: a(n) = 20*a(n-1) -64*a(n-2) for n>3
k=3: a(n) = 64*a(n-1) -960*a(n-2) +4096*a(n-3) for n>4
k=4: [order 5] for n>6
k=5: [order 10] for n>11
k=6: [order 22] for n>23

A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 4, 10, 11, 0, 8, 36, 48, 49, 0, 16, 136, 236, 256, 257, 0, 32, 528, 1248, 1508, 1538, 1539, 0, 64, 2080, 6896, 9696, 10256, 10298, 10299, 0, 128, 8256, 39168, 66384, 74784, 75848, 75904, 75905, 0, 256, 32896, 226496, 475136, 586352, 607520, 609368, 609440, 609441
Offset: 0

Views

Author

Mark Wildon, Aug 13 2015

Keywords

Comments

C_t(n) is the number of sequences of t top-to-random shuffles that leave a deck of n cards invariant, if each shuffle is permitted to flip the orientation of the card it moves.
C_t(n) = where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.
C_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks.
C_t(n) = C_t(t) if n > t.

Examples

			Triangle starts:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   10,   11;
  0,  8,   36,   48,   49;
  0, 16,  136,  236,  256,   257;
  0, 32,  528, 1248, 1508,  1538,  1539;
  0, 64, 2080, 6896, 9696, 10256, 10298, 10299;
  ...

Links

Crossrefs

Columns n=0,1,2,3 give A000007, A000079, A007582, A233162 (proved for n=3 in reference above).
Main diagonal gives A004211.
Cf. A075497.

Programs

  • Maple
    with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 13 2015
  • Mathematica
    CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];
Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Formula

G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).
C_t(n) = Sum_{i=0..n} A075497(t,i).
Showing 1-6 of 6 results.

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