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 17 results. Next

A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 50, 25, 6, 22, 86, 130, 95, 36, 7, 29, 148, 296, 295, 161, 49, 8, 37, 239, 610, 791, 581, 252, 64, 9, 46, 367, 1163, 1897, 1792, 1036, 372, 81, 10, 56, 541, 2083, 4166, 4900, 3612, 1716, 525, 100, 11, 67, 771, 3544, 8518, 12174, 11088, 6672, 2685, 715, 121, 12
Offset: 1

Views

Author

R. H. Hardin, Apr 23 2013

Keywords

Comments

Table starts
..2...4...7...11....16.....22.....29......37......46.......56.......67
..3...9..22...46....86....148....239.....367.....541......771.....1068
..4..16..50..130...296....610...1163....2083....3544.....5776.....9076
..5..25..95..295...791...1897...4166....8518...16414....30086....52834
..6..36.161..581..1792...4900..12174...27966...60172...122464...237590
..7..49.252.1036..3612..11088..30738...78354..186142...416394...884236
..8..64.372.1716..6672..22716..69498..194634..505912..1233584..2845492
..9..81.525.2685.11517..43065.144111..439791.1241383..3276559..8157227
.10.100.715.4015.18832..76714.278707..920491.2803658..7963384.21280337
.11.121.946.5786.29458.129844.508937.1808521.5911763.17978389.51325352
From Charles A. Lane, Aug 22 2013: (Start)
The first column is also the coefficients of a in y''[x] - a*x^n*y[x] + b*en*y[x] = 0 where n = 0. The recursion yields coefficients of a, a*b*en, a*b^2*en^2 etc.
The second column is obtained when n=1, the third column when n=2. The final column is for n=10.
Example: Write a normal recursion for n=4. For convenience set x to 1. Running the recursion yields
1-(b en)/2+(b^2 en^2)/24+1/30 (a-(b^3 en^3)/24)+(-384 a b en+b^4 en^4)/40320+(2064 a b^2 en^2-b^5 en^5)/3628800+(120960 a^2-7104 a b^3 en^3+b^6 en^6)/479001600+(-4682880 a^2 b en+18984 a b^4 en^4-b^7 en^7)/87178291200+(54268416 a^2 b^2 en^2-43008 a b^5 en^5+b^8 en^8)/20922789888000.
The coefficient of a is 24, the coefficient of a b en is 384 and the coefficient of a b^2 en^2 is 2064. Dividing by 4! yields a sequence of 1,16,86... , the same as column 5 without the leading 1. There is a hint of unity among the oscillators. (End)

Examples

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

Links

Crossrefs

Column 2 is A000290(n+1).
Column 3 is A002412(n+1).
Column 4 is A006324(n+1).
Row 1 is A000124.
Row 2 is A223718.
Row 3 is A223659.
Cf. A071920, A071921 (larger and reflected versions of table). - Alois P. Heinz, Sep 22 2013

Programs

  • Maple
    T:= (n, k)-> add(binomial(k+2*j-1, 2*j), j=0..n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 22 2013
  • Mathematica
    T[n_, k_] := Sum[Binomial[k + 2*j - 1, 2*j], {j, 0, n}]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

Formula

Empirical: columns k=1..7 are polynomials of degree k.
Empirical: rows n=1..7 are polynomials of degree 2n.
T(n,k) = Sum_{j=0..n} C(k+2*j-1,2*j). - Alois P. Heinz, Sep 22 2013

A223864 T(n,k)=Number of nXk 0..3 arrays with rows and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 4884, 1225, 56, 610, 14751, 41682, 24199, 3136, 84, 1163, 52591, 273959, 315124, 93731, 7056, 120, 2083, 165212, 1477240, 3017129, 1771012, 303560, 14400, 165, 3544, 468292, 6818350, 22852913
Offset: 1

Views

Author

R. H. Hardin Mar 28 2013

Keywords

Comments

Table starts
...4....16.......50.......130.........296...........610...........1163
..10...100......684......3526.......14751.........52591.........165212
..20...400.....4884.....41682......273959.......1477240........6818350
..35..1225....24199....315124.....3017129......22852913......144081276
..56..3136....93731...1771012....23738426.....243933798.....2030417942
..84..7056...303560...8008548...145947740....1989679315....21476594002
.120.14400...857696..30627033...740441932...13140481520...181330154458
.165.27225..2175884.102479569..3217594840...73068868012..1271807435844
.220.48400..5058530.307435001.12305144319..352040804450..7630189031428
.286.81796.10940664.842078930.42270004211.1502130487437.40055722078772

Examples

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

Links

Crossrefs

Column 1 is A000292(n+1)
Column 2 is A001249
Row 1 is A223659

Formula

Empirical: columns k=1..7 are polynomials of degree 3*k for n>0,0,0,1,2,4,6
Empirical: rows n=1..7 are polynomials of degree 6*n

A224173 T(n,k) = number of n X k 0..3 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 4739, 1225, 56, 610, 14751, 38561, 22988, 3136, 84, 1163, 52591, 242114, 272130, 87878, 7056, 120, 2083, 165212, 1253770, 2335459, 1460836, 282372, 14400, 165, 3544, 468292, 5588411, 15925611
Offset: 1

Views

Author

R. H. Hardin, Mar 31 2013

Keywords

Examples

			Table starts:
    4    16       50       130         296          610           1163
   10   100      684      3526       14751        52591         165212
   20   400     4739     38561      242114      1253770        5588411
   35  1225    22988    272130     2335459     15925611       91494280
   56  3136    87878   1460836    16625026    143558572     1012166273
   84  7056   282372   6425876    95808564   1038484760     8857798353
  120 14400   794220  24197608   468021427   6360047093    65713691148
  165 27225  2010035  80350989  1994287334  33901838632   426013124302
  220 48400  4668304 240416852  7568051210 160168789130  2451904991177
  286 81796 10095924 658890738 25994968917 680269560125 12667946702827
  ...
Some solutions for n=3 k=4
..0..0..1..0....0..0..1..2....0..0..3..0....0..2..0..0....0..3..3..1
..1..3..3..1....0..1..3..2....3..3..3..1....1..2..0..0....1..3..3..1
..1..3..3..3....0..3..3..2....3..3..3..2....2..2..1..0....1..3..3..3

Links

Crossrefs

Main diagonal is A224167.
Columns 1..7 are A000292(n+1), A001249, A224168, A224169, A224170, A224171, A224172.
Rows 1..7 are A223659, A223865, A224174, A224175, A224176, A224177, A224178.
Cf. A223838.

Formula

Empirical: columns k=1..7 are polynomials of degree 3*k for n>0,0,0,3,6,9,12.
Empirical: rows n=1..5 are polynomials of degree 6*n for k>0,0,0,2,6.

Extensions

Name corrected by Andrew Howroyd, Mar 18 2025

A223801 T(n,k)=Number of nXk 0..3 arrays with rows and antidiagonals unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 64, 130, 2500, 4096, 256, 296, 16900, 110116, 65536, 1024, 610, 87616, 1658703, 4816168, 1048576, 4096, 1163, 372100, 16979881, 151310069, 210163664, 16777216, 16384, 2083, 1352569, 131295500, 2844578252, 13602542576
Offset: 1

Views

Author

R. H. Hardin Mar 27 2013

Keywords

Comments

Table starts
.......4............16................50..................130
......16...........256..............2500................16900
......64..........4096............110116..............1658703
.....256.........65536...........4816168............151310069
....1024.......1048576.........210163664..........13602542576
....4096......16777216........9169032476........1216562667529
...16384.....268435456......400006582368......108631485025292
...65536....4294967296....17450517286804.....9695922803812530
..262144...68719476736...761287955888788...865293308203272685
.1048576.1099511627776.33211580867804324.77218182866179219113

Examples

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

Links

Crossrefs

Column 1 is A000302
Column 2 is A001025
Row 1 is A223659
Row 2 is A223756

Formula

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 16*a(n-1)
k=3: [order 10]
k=4: [order 27]
Empirical: rows n=1..6 are polynomials of degree 6*n for k>0,0,0,0,2,3

A223987 T(n,k)=Number of nXk 0..3 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 5029, 1225, 56, 610, 14751, 44803, 25410, 3136, 84, 1163, 52591, 308470, 358118, 99634, 7056, 120, 2083, 165212, 1738756, 3770722, 2086196, 325120, 14400, 165, 3544, 468292, 8350154, 31585056
Offset: 1

Views

Author

R. H. Hardin Mar 30 2013

Keywords

Comments

Table starts
...4....16.......50........130.........296...........610...........1163
..10...100......684.......3526.......14751.........52591.........165212
..20...400.....5029......44803......308470.......1738756........8350154
..35..1225....25410.....358118.....3770722......31585056......219861244
..56..3136....99634....2086196....31831914.....378122264.....3661410444
..84..7056...325120....9647292...204647416....3322756326....43307637038
.120.14400...922768...37395816..1067023886...22985966340...392525216516
.165.27225..2346883..126087157..4710529013..131366850521..2873859236297
.220.48400..5462600..379654704.18159308422..642224541548.17659521902693
.286.81796.11818092.1040942916.62548820489.2756467192963.93729371629362

Examples

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

Links

Crossrefs

Column 1 is A000292(n+1)
Column 2 is A001249
Row 1 is A223659
Row 2 is A223865

Formula

Empirical: columns k=1..7 are polynomials of degree 3*k
Empirical: rows n=1..7 are polynomials of degree 6*n

A224204 T(n,k)=Number of nXk 0..3 arrays with rows unimodal and antidiagonals nondecreasing.

Original entry on oeis.org

4, 16, 16, 50, 160, 64, 130, 984, 1600, 256, 296, 4580, 16060, 16000, 1024, 610, 17723, 108625, 263516, 160000, 4096, 1163, 59792, 586343, 2411246, 4357084, 1600000, 16384, 2083, 180821, 2734683, 16355242, 54177872, 72105068, 16000000, 65536
Offset: 1

Views

Author

R. H. Hardin Apr 01 2013

Keywords

Comments

Table starts
.......4..........16............50.............130..............296
......16.........160...........984............4580............17723
......64........1600.........16060..........108625...........586343
.....256.......16000........263516.........2411246.........16355242
....1024......160000.......4357084........54177872........451319098
....4096.....1600000......72105068......1229044416......12652618110
...16384....16000000....1193130640.....27957232796.....357890479324
...65536...160000000...19742052632....636184842092...10153767871028
..262144..1600000000..326659600368..14476260508500..288290902851198
.1048576.16000000000.5405039750704.329391607167600.8186391229197618

Examples

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

Links

Crossrefs

Column 1 is A000302
Column 2 is 16*10^(n-1)
Row 1 is A223659
Row 2 is A224058

Formula

Empirical: columns k=1..7 have recurrences of order 1,1,7,10,19,25,41 for n>0,0,0,12,23,32,48
Empirical: rows n=1..7 are polynomials of degree 6*n for k>0,0,0,2,3,4,5

A224281 T(n,k)=Number of nXk 0..3 arrays with diagonals and rows unimodal and antidiagonals nondecreasing.

Original entry on oeis.org

4, 16, 16, 50, 160, 64, 130, 984, 1600, 256, 296, 4580, 13683, 16000, 1024, 610, 17723, 84132, 186516, 160000, 4096, 1163, 59792, 442089, 1334973, 2596992, 1600000, 16384, 2083, 180821, 2059793, 8073038, 21348990, 37128051, 16000000, 65536, 3544
Offset: 1

Views

Author

R. H. Hardin Apr 02 2013

Keywords

Comments

Table starts
.......4..........16............50............130.............296
......16.........160...........984...........4580...........17723
......64........1600.........13683..........84132..........442089
.....256.......16000........186516........1334973.........8073038
....1024......160000.......2596992.......21348990.......137489538
....4096.....1600000......37128051......356222482......2425304290
...16384....16000000.....537465766.....6172817040.....45275725025
...65536...160000000....7804602744...109166159263....883012703273
..262144..1600000000..113382138975..1947747629183..17667432461262
.1048576.16000000000.1646661944858.34864494529806.358042017265316

Examples

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

Links

Crossrefs

Column 1 is A000302
Column 2 is 16*10^(n-1)
Row 1 is A223659
Row 2 is A224058

Formula

Empirical: columns k=1..4 have recurrences of order 1,1,25,52 for n>0,0,26,59
Empirical: rows n=1..6 are polynomials of degree 6*n for k>0,0,3,7,11,15

A223876 T(n,k)=Number of nXk 0..3 arrays with rows, diagonals and antidiagonals unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 64, 130, 2500, 4096, 256, 296, 16900, 99223, 65536, 1024, 610, 87616, 1336985, 3863372, 1048576, 4096, 1163, 372100, 12520369, 88682677, 152918517, 16777216, 16384, 2083, 1352569, 90648289, 1271992512, 5941888105
Offset: 1

Views

Author

R. H. Hardin Mar 28 2013

Keywords

Comments

Table starts
.......4............16................50..................130
......16...........256..............2500................16900
......64..........4096.............99223..............1336985
.....256.........65536...........3863372.............88682677
....1024.......1048576.........152918517...........5941888105
....4096......16777216........6066668157.........411716468431
...16384.....268435456......240345697904.......28928809433978
...65536....4294967296.....9519219712534.....2033941972287214
..262144...68719476736...377068749332794...142745781634483746
.1048576.1099511627776.14936662560715369.10010372252279889400

Examples

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

Links

Crossrefs

Column 1 is A000302
Column 2 is A001025
Row 1 is A223659
Row 2 is A223756

Formula

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 16*a(n-1)
k=3: [recurrence of order 28]
Empirical: rows n=1..4 are polynomials of degree 6*n for k>0,0,1,10

A223762 T(n,k)=Number of nXk 0..3 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 50, 130, 2500, 2500, 130, 296, 16900, 61733, 16900, 296, 610, 87616, 916107, 916107, 87616, 610, 1163, 372100, 9478535, 26631193, 9478535, 372100, 1163, 2083, 1352569, 74824917, 499583168, 499583168, 74824917, 1352569, 2083
Offset: 1

Views

Author

R. H. Hardin Mar 27 2013

Keywords

Comments

Table starts
....4.......16..........50............130..............296.................610
...16......256........2500..........16900............87616..............372100
...50.....2500.......61733.........916107..........9478535............74824917
..130....16900......916107.......26631193........499583168..........6754232986
..296....87616.....9478535......499583168......15947472102........350182483445
..610...372100....74824917.....6754232986.....350182483445......12025063773557
.1163..1352569...477860225....70657105931....5733827943118.....298630624023222
.2083..4338889..2571238699...600526842770...73972033945807....5692159574625373
.3544.12559936.12006271464..4295532642860..782389879664731...86933612444360250
.5776.33362176.49749360288.26553802745045.6990377642784235.1098959847553820404

Examples

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

Links

Crossrefs

Column 1 is A223659

Formula

Empirical: columns k=1..5 are polynomials of degree 6*k for n>0,0,0,2,5

A223850 T(n,k)=Number of nXk 0..3 arrays with rows and columns unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 50, 130, 2500, 2500, 130, 296, 16900, 64660, 16900, 296, 610, 87616, 1006318, 1006318, 87616, 610, 1163, 372100, 10804883, 32464394, 10804883, 372100, 1163, 2083, 1352569, 87613063, 664770145, 664770145, 87613063, 1352569, 2083
Offset: 1

Views

Author

R. H. Hardin Mar 27 2013

Keywords

Comments

Table starts
....4.......16..........50............130...............296................610
...16......256........2500..........16900.............87616.............372100
...50.....2500.......64660........1006318..........10804883...........87613063
..130....16900.....1006318.......32464394.........664770145.........9625018705
..296....87616....10804883......664770145.......24291817048.......594861333098
..610...372100....87613063.....9625018705......594861333098.....23571122875874
.1163..1352569...570145144...106061178908....10609537390768....655695544507798
.2083..4338889..3107412546...936977517660...146233793223364..13664368909406118
.3544.12559936.14632983606..6894239747078..1627190607796056.223826020405698042
.5776.33362176.60951077586.43487182160312.15110227641526318

Examples

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

Links

Crossrefs

Column 1 is A223659
Column 2 is A223756

Formula

Empirical: columns k=1..5 are polynomials of degree 6*k
Showing 1-10 of 17 results. Next

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