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 13 results. Next

A264285 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,0 0,1 1,0 or -1,-2.

Original entry on oeis.org

1, 4, 1, 8, 10, 1, 16, 32, 26, 1, 33, 102, 132, 69, 1, 69, 360, 675, 556, 181, 1, 145, 1228, 4189, 4484, 2324, 476, 1, 300, 4156, 23852, 47492, 29742, 9724, 1252, 1, 624, 14148, 134432, 448821, 537057, 197283, 40692, 3292, 1, 1300, 48188, 768664, 4227024
Offset: 1

Views

Author

R. H. Hardin, Nov 10 2015

Keywords

Comments

Table starts
.1.....4.......8........16..........33.............69..............145
.1....10......32.......102.........360...........1228.............4156
.1....26.....132.......675........4189..........23852...........134432
.1....69.....556......4484.......47492.........448821..........4227024
.1...181....2324.....29742......537057........8405669........131452948
.1...476....9724....197283.....6080234......157344756.......4076914388
.1..1252...40692...1308629....68815948.....2943284092.....126311779972
.1..3292..170268...8680430...778858184....55051679668....3911932445892
.1..8657..712468..57579243..8815152033..1029653214581..121136137544916
.1.22765.2981244.381936079.99770013733.19257696830753.3750881659750212

Examples

			Some solutions for n=4 k=4
..7..0..9..2..3....7..1..9..2..4....0..1..9..2..3....0..1..9..2..3
.12..1..6..8..4....0.13..6..3..8....5.13..6..7..4...12..5.14..7..4
..5.11.19.13.14....5.10.11.12.14...10.11.12..8.14...10..6.11..8.13
.10.23.16.17.18...22.15.16.18.19...22.15.24.17.18...15.23.16.17.19
.15.20.21.22.24...20.21.17.23.24...20.16.21.23.19...20.21.22.18.24

Links

Crossrefs

Column 2 is A099234(n+1).
Row 1 is A264166.

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +3*a(n-2) +3*a(n-3) +a(n-4)
k=3: a(n) = 3*a(n-1) +4*a(n-2) +4*a(n-3)
k=4: a(n) = 7*a(n-1) -2*a(n-2) -2*a(n-3) -6*a(n-4) +a(n-5) +3*a(n-6)
k=5: [order 28]
k=6: [order 36]
k=7: [order 34]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2) +2*a(n-3) +a(n-4) +a(n-5) -a(n-6)
n=2: a(n) = a(n-1) +4*a(n-2) +10*a(n-3) +12*a(n-4) +8*a(n-5) for n>7
n=3: [order 56]

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 8, 1, 1, 1, 6, 15, 26, 28, 13, 1, 1, 1, 7, 21, 45, 69, 60, 21, 1, 1, 1, 8, 28, 71, 140, 181, 129, 34, 1, 1, 1, 9, 36, 105, 251, 431, 476, 277, 55, 1, 1, 1, 10, 45, 148, 413, 882, 1326, 1252, 595, 89, 1
Offset: 0

Views

Author

Paul Barry, Oct 08 2004

Keywords

Examples

			Rows begin
  1, 1, 1,  1,  1,   1, ...
  1, 1, 2,  3,  5,   8, ...
  1, 1, 3,  6, 13,  28, ...
  1, 1, 4, 10, 26,  69, ...
  1, 1, 5, 15, 45, 140, ...
Row 1 is the 0-section of 1/(1-x-x)   (A000079);
Row 2 is the 1-section of 1/(1-x-x^2) (A000045);
Row 3 is the 2-section of 1/(1-x-x^3) (A000930);
Row 4 is the 3-section of 1/(1-x-x^4) (A003269);
etc.

Links

Crossrefs

Sums of antidiagonals are A099236.
Columns include A000217, A008778.
Rows include A000045, A002478, A099234, A099235.
Main diagonal gives A099237.
Cf. A099238.

Formula

Square array T(n, k) = Sum_{j=0..n} binomial(k(n-j), j).
Rows are generated by 1/(1-x(1+x)^k) and satisfy a(n) = Sum_{k=0..n} binomial(n, k)a(n-k-1).

A200251 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its previous elements modulo (k+1).

Original entry on oeis.org

2, 3, 3, 4, 6, 5, 5, 10, 12, 8, 6, 15, 26, 24, 13, 7, 21, 45, 69, 48, 21, 8, 28, 75, 135, 181, 96, 34, 9, 36, 112, 267, 405, 476, 192, 55, 10, 45, 164, 448, 951, 1215, 1252, 384, 89, 11, 55, 225, 750, 1792, 3387, 3645, 3292, 768, 144, 12, 66, 305, 1125, 3434, 7168, 12063
Offset: 1

Views

Author

R. H. Hardin Nov 15 2011

Keywords

Comments

Table starts
...2....3.....4.....5......6.......7.......8........9.......10........11
...3....6....10....15.....21......28......36.......45.......55........66
...5...12....26....45.....75.....112.....164......225......305.......396
...8...24....69...135....267.....448.....750.....1125.....1690......2376
..13...48...181...405....951....1792....3434.....5625.....9365.....14256
..21...96...476..1215...3387....7168...15724....28125....51895.....85536
..34..192..1252..3645..12063...28672...71970...140625...287570....513216
..55..384..3292.10935..42963..114688..329455...703125..1593535...3079296
..89..768..8657.32805.153015..458752.1508139..3515625..8830385..18475776
.144.1536.22765.98415.544971.1835008.6903702.17578125.48932530.110854656

Examples

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

Links

Crossrefs

Column 1 is A000045(n+2)
Column 2 is A003945
Column 3 is A099234(n+1)
Column 4 is A005030(n-1)
Column 6 is A002042(n-1)
Row 2 is A000217(n+1)

Formula

Empirical: T(n,2k) = (2*k+1)*(k+1)^(n-1)

A360076 a(n) = Sum_{k=0..n} binomial(3*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 5, 20, 90, 430, 2136, 10937, 57307, 305822, 1656482, 9083432, 50328114, 281324294, 1584578746, 8984740485, 51242962251, 293772468164, 1691974930584, 9785378133297, 56805049768157, 330880419984832, 1933299689139364, 11328101469158554
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Cf. A052709, A073155, A360082, A360083.
Cf. A000108, A099234.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^3)))

Formula

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^3 * A(x)).
G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^3 )).

A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.

Original entry on oeis.org

1, 3, 15, 55, 198, 681, 2263, 7341, 23331, 72928, 224814, 684882, 2065346, 6173466, 18310212, 53935350, 157904130, 459755694, 1332010954, 3841812480, 11035346151, 31579747613, 90061069065, 256028590665, 725715896698, 2051465107719, 5784472106577, 16271956316851
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2025

Keywords

Links

Crossrefs

Column k=3 of A362125.
Cf. A099234, A382613.
Cf. A000217, A001628, A382406.
Cf. A382615.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 02 2025
  • Mathematica
    Table[Sum[Binomial[k+2,2]*Binomial[3*k,n-k],{k,0,n}],{n,0,27}] (* Vincenzo Librandi, Apr 02 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(3*k, n-k));

Formula

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(3*k,n-k).
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) - 33*a(n-4) - 24*a(n-5) + 39*a(n-6) + 108*a(n-7) + 123*a(n-8) + 84*a(n-9) + 36*a(n-10) + 9*a(n-11) + a(n-12).
G.f.: -1/(x^4+3*x^3+3x^2+x-1)^3. - Vincenzo Librandi, Apr 02 2025

A361842 Expansion of 1/(1 - 9*x*(1+x)^3)^(1/3).

Original entry on oeis.org

1, 3, 27, 243, 2352, 23607, 242757, 2539431, 26904492, 287858421, 3104029755, 33684914907, 367483636746, 4026930734223, 44295829667055, 488855016668727, 5410588668898995, 60035381850523284, 667643481187840206, 7439651232903588528, 83050643822779921347
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Links

Crossrefs

Column k=3 of A361839.
Cf. A099234, A361812, A361845.

Programs

  • Mathematica
    a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, -2^8/3^5]; Array[a,21,0] (* Stefano Spezia, Jul 11 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^3)^(1/3))

Formula

n*a(n) = 3 * ( (3*n-2)*a(n-1) + 3*(3*n-4)*a(n-2) + 3*(3*n-6)*a(n-3) + (3*n-8)*a(n-4) ) for n > 3.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(3*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 2/3-n, 2/3-n], -2^8/3^5). - Stefano Spezia, Jul 11 2024

A116089 Riordan array (1, x*(1+x)^3).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 1, 15, 9, 1, 0, 0, 20, 36, 12, 1, 0, 0, 15, 84, 66, 15, 1, 0, 0, 6, 126, 220, 105, 18, 1, 0, 0, 1, 126, 495, 455, 153, 21, 1, 0, 0, 0, 84, 792, 1365, 816, 210, 24, 1, 0, 0, 0, 36, 924, 3003, 3060, 1330, 276, 27, 1
Offset: 0

Views

Author

Paul Barry, Feb 04 2006

Keywords

Examples

			Triangle begins as:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 1, 15,   9,   1;
  0, 0, 20,  36,  12,    1;
  0, 0, 15,  84,  66,   15,    1;
  0, 0,  6, 126, 220,  105,   18,    1;
  0, 0,  1, 126, 495,  455,  153,   21,   1;
  0, 0,  0,  84, 792, 1365,  816,  210,  24,  1;
  0, 0,  0,  36, 924, 3003, 3060, 1330, 276, 27, 1;

Links

Crossrefs

Row sums are A099234. Diagonal sums are A116090.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(3*k, n-k) ))); # G. C. Greubel, May 09 2019
  • Magma
    [[Binomial(3*k, n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019
  • Mathematica
    Flatten[Table[Binomial[3k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Feb 05 2012 *)
  • PARI
    {T(n,k) = binomial(3*k, n-k)}; \\ G. C. Greubel, May 09 2019
  • Sage
    [[binomial(3*k, n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019

Formula

G.f.: 1/(1-x*y*(1+x)^3).
Number triangle T(n,k) = C(3*k,n-k) = C(n,k)*C(4*k,n)/C(4*k,k).

A382613 Expansion of 1/(1 - x*(1 + x)^3)^2.

Original entry on oeis.org

1, 2, 9, 28, 88, 270, 808, 2386, 6960, 20104, 57607, 163950, 463907, 1306104, 3661248, 10223820, 28452400, 78941412, 218426608, 602886704, 1660329597, 4563175466, 12517834605, 34280427828, 93729509848, 255900484218, 697712467704, 1899912606358, 5167488465184
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2025

Keywords

Links

Crossrefs

Cf. A099234, A382614.
Cf. A001629, A362126.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 08 2025
  • Mathematica
    Table[Sum[(k+1)*Binomial[3*k,n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, Apr 08 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*binomial(3*k, n-k));

Formula

a(n) = Sum_{k=0..n} (k+1) * binomial(3*k,n-k).
a(n) = 2*a(n-1) + 5*a(n-2) - 13*a(n-4) - 20*a(n-5) - 15*a(n-6) - 6*a(n-7) - a(n-8).
G.f.: 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2. - Vincenzo Librandi, Apr 08 2025

A360087 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k,n-k).

Original entry on oeis.org

1, -1, -2, 2, 6, -5, -17, 12, 48, -28, -135, 63, 378, -134, -1054, 259, 2927, -408, -8096, 280, 22305, 1551, -61210, -10638, 167310, 46683, -455489, -175852, 1234960, 612380, -3334215, -2031953, 8962498, 6523626, -23981046, -20445373, 63855135, 62900496
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Links

Crossrefs

Cf. A077979, A360088, A360089.
Cf. A099234.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(3*k, n-k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1+x*(1+x)^3))

Formula

a(n) = -a(n-1) - 3*a(n-2) - 3*a(n-3) - a(n-4).
G.f.: 1/(1 + x*(1+x)^3).

A360090 a(n) = Sum_{k=0..n} binomial(5*k,n-k).

Original entry on oeis.org

1, 1, 6, 21, 71, 251, 882, 3088, 10829, 37975, 133146, 466852, 1636944, 5739647, 20125051, 70564951, 247423522, 867546829, 3041899638, 10665883415, 37398034921, 131129599227, 459782762029, 1612146986543, 5652708454881, 19820223058176, 69496108849357
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Comments

The number of ways to place non-overlapping Young diagrams of shape (2,1,1,1,1) on an 9 by n rectangle. - Per Alexandersson, Jul 01 2025

Links

Crossrefs

Cf. A002478, A099234, A099235.

Programs

  • Maple
    seq(add(binomial(5*k,n-k),k=0..n), n=0..50); # Robert Israel, Jul 09 2025
  • PARI
    a(n) = sum(k=0, n, binomial(5*k, n-k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-x*(1+x)^5))

Formula

a(n) = a(n-1) + 5*a(n-2) + 10*a(n-3) + 10*a(n-4) + 5*a(n-5) + a(n-6).
G.f.: 1/(1 - x*(1+x)^5).
Showing 1-10 of 13 results. Next

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