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 11 results. Next

A195644 T(n,k) is the number of lower triangles of an n X n 0..k array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 15, 1, 1, 1, 3, 19, 199, 1, 1, 1, 3, 19, 379, 6247, 1, 1, 1, 3, 19, 391, 22506, 505623, 1, 1, 1, 3, 19, 391, 25428, 4063437, 105997283, 1, 1, 1, 3, 19, 391, 25532, 5422820, 2303397986, 58923059879, 1, 1, 1, 3, 19, 391, 25532, 5536654
Offset: 1

Views

Author

R. H. Hardin, Sep 21 2011

Keywords

Examples

			Table starts:
  1         1          1          1          1          1 ...
  1         1          1          1          1          1 ...
  1         3          3          3          3          3 ...
  1        15         19         19         19         19 ...
  1       199        379        391        391        391 ...
  1      6247      22506      25428      25532      25532 ...
  1    505623    4063437    5422820    5536654    5539434 ...
  1 105997283 2303397986 3868544673 4102276124 4116036800 ...
  ...
Some solutions for n=5, k=4:
..1..........1..........1..........1..........1..........1..........1
..1.1........1.1........2.0........2.0........2.0........2.0........2.0
..3.0.0......3.0.0......0.1.2......2.1.0......0.1.2......2.0.1......1.2.0
..0.1.1.2....0.1.1.2....1.1.0.2....0.1.1.2....1.1.0.2....0.0.2.2....1.2.1.0
..0.2.2.0.1..0.2.2.0.1..1.2.1.0.1..0.2.2.0.1..1.2.1.0.1..0.4.0.0.1..0.0.2.2.1

Links

Crossrefs

Columns 1..7 are A000012, A195515, A195639, A195640, A195641, A195642, A195643.
Main diagonal is A195638.
Cf. A195522, A257493.

Programs

  • PARI
    \\ adapted from program for A257493.
T(n, k)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, min(k, h-i), for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1] + x^(r-1), 0, src[i, 2], r))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, May 16 2020

Formula

T(n,k) = T(n,k-1) for k >= n, n >= 2. - Andrew Howroyd, May 16 2020

A195515 Number of lower triangles of an n X n -1..1 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 3, 15, 199, 6247, 505623, 105997283, 58923059879, 87882127554983, 355824020816195749, 3945677945542866439849, 120747614641791880577007593, 10263027079252827219477164325361, 2436143827877272419846561427146618539, 1622665432590314350761059585434700996826279
Offset: 1

Views

Author

R. H. Hardin, Sep 20 2011

Keywords

Examples

			Some solutions for 4 X 4:
..0........0........0........0........0........0........0........0
..1-1.....-1.1......1-1......0.0.....-1.1......0.0.....-1.1......1-1
..0.0.0....1-1.0....0.0.0....0.0.0....0.0.0....1.0-1....0.0.0...-1.1.0
.-1.1.0.0..0.0.0.0.-1.1.0.0..0.0.0.0..1-1.0.0.-1.0.1.0..1-1.0.0..0.0.0.0

Crossrefs

Column 1 of A195522 and column 2 of A195644.

Extensions

a(13)-a(16) from Andrew Howroyd, May 16 2020

A195523 Number of lower triangles of a 5 X 5 -n..n array with all row and column sums zero.

Original entry on oeis.org

199, 3753, 27267, 121367, 401565, 1089411, 2563933, 5423365, 10557163, 19228309, 33165903, 54668043, 86714993, 133092639, 198526233, 288824425, 411033583, 573602401, 786556795, 1061685087, 1412733477, 1855611803, 2408609589
Offset: 1

Views

Author

R. H. Hardin, Sep 20 2011

Keywords

Comments

Row 5 of A195522.

Examples

			Some solutions for n=4:
..0..........0..........0..........0..........0..........0..........0
..0.0.......-2.2.......-3.3........0.0.......-3.3........3-3........3-3
..0-3.3.....-2.4-2.....-2-1.3......2-2.0......0-1.1.....-3.0.3......2.0-2
..0.3.0-3....2-2.4-4....4-1-4.1....0-1.2-1....1.0-1.0....4-1.1-4...-4-1.4.1
..0.0-3.3.0..2-4-2.4.0..1-1.1-1.0.-2.3-2.1.0..2-2.0.0.0.-4.4-4.4.0.-1.4-2-1.0

Links

Crossrefs

Cf. A195522.

Formula

Empirical: a(n) = (643/45)*n^6 + (643/15)*n^5 + (2165/36)*n^4 + (293/6)*n^3 + (4423/180)*n^2 + (73/10)*n + 1.
Conjectures from Colin Barker, May 08 2018: (Start)
G.f.: x*(199 + 2360*x + 5175*x^2 + 2346*x^3 + 213*x^4 - 6*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A195524 Number of lower triangles of a 6X6 -n..n array with all row and column sums zero.

Original entry on oeis.org

6247, 860017, 23663523, 286168923, 2106810049, 11131321791, 46387885537, 161770028369, 491117662815, 1334411256817, 3311007744155, 7616624997827, 16434220066785, 33565300493927, 65367853464305, 122110269851089
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Row 6 of A195522

Examples

			Some solutions for n=4
..0............0............0............0............0............0
..2-2.........-1.1.........-1.1.........-4.4..........0.0..........0.0
..3-4.1.......-1.3-2.......-1-3.4........1-2.1........0-1.1.......-2.1.1
.-1.3-4.2.....-2.4.2-4......2-3.2-1......1.3.0-4.....-2.2.3-3.....-2.4.0-2
.-1.3-1-2.1....1-4.4.2-3....3.4-3-3-1...-2-1.1.3-1...-1-1.0.1.1....1-1-2.0.2
.-3.0.4.0-1.0..3-4-4.2.3.0.-3.1-3.4.1.0..4-4-2.1.1.0..3.0-4.2-1.0..3-4.1.2-2.0

Links

Formula

Empirical: a(n) = (7389349/90720)*n^10 + (7389349/18144)*n^9 + (836251/864)*n^8 + (4318165/3024)*n^7 + (6254923/4320)*n^6 + (4563293/4320)*n^5 + (10247161/18144)*n^4 + (249983/1134)*n^3 + (21959/360)*n^2 + (3469/315)*n + 1

A195525 Number of lower triangles of a 7X7 -n..n array with all row and column sums zero.

Original entry on oeis.org

505623, 839301197, 122092290831, 5151383545261, 103074513713971, 1252997799222737, 10665939121760907, 69488770600802237, 367679036207885407, 1647079991451268941, 6438544930173240123, 22466897655527435065
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Row 7 of A195522

Examples

			Some solutions for n=4
..0..............0..............0..............0..............0
.-4.4...........-4.4............3-3...........-2.2...........-4.4
..1.2-3..........2-3.1..........3-2-1.........-2.1.1.........-2.0.2
..4-1-1-2.......-3-3.3.3........2-1.1-2.......-4-1.1.4........4.3-3-4
.-1-1.2.2-2......3-3-1-1.2.....-2.2.1.3-4......3-2.2.0-3.....-1-4-3.4.4
.-2-3.2-1.3.1...-1.3-3.0-3.4...-2.3.2-4.4-3....2-4.0-1.2.1....1.0.0.3-3-1
..2-1.0.1-1-1.0..3.2.0-2.1-4.0.-4.1-3.3.0.3.0..3.4-4-3.1-1.0..2-3.4-3-1.1.0

Links

A195516 Number of lower triangles of an n X n -2..2 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 5, 65, 3753, 860017, 839301197, 3535646416019, 65638445025610187, 5443595184898837903553
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Column 2 of A195522

Examples

			Some solutions for n=4
..0........0........0........0........0........0........0........0
.-1.1.....-2.2.....-2.2......1-1......1-1......0.0......2-2......1-1
..0-2.2....1.0-1....2.0-2...-1-1.2....1-1.0...-1.2-1....0.2-2....0.0.0
..1.1-2.0..1-2.1.0..0-2.2.0..0.2-2.0.-2.2.0.0..1-2.1.0.-2.0.2.0.-1.1.0.0

A195517 Number of lower triangles of an n X n -3..3 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 7, 175, 27267, 23663523, 122092290831, 3806404472713603, 731708552185212577507
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Column 3 of A195522

Examples

			Some solutions for n=4
..0........0........0........0........0........0........0........0
.-1.1......0.0......2-2......0.0.....-3.3.....-2.2.....-2.2......2-2
.-2-1.3....2-2.0....1.0-1....3.0-3....3-3.0....0-3.3....0-3.3....0.1-1
..3.0-3.0.-2.2.0.0.-3.2.1.0.-3.0.3.0..0.0.0.0..2.1-3.0..2.1-3.0.-2.1.1.0

A195518 Number of lower triangles of an n X n -4..4 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 9, 369, 121367, 286168923, 5151383545261, 720374496860187931, 798423610752730381095001
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Column 4 of A195522

Examples

			Some solutions for n=4
..0........0........0........0........0........0........0........0
..3-3......4-4......3-3.....-3.3.....-3.3.....-1.1.....-2.2......1-1
.-3.4-1....0.0.0...-3.0.3....2-1-1....2-3.1....0-4.4...-2.1.1....1.0-1
..0-1.1.0.-4.4.0.0..0.3-3.0..1-2.1.0..1.0-1.0..1.3-4.0..4-3-1.0.-2.1.1.0

A195519 Number of lower triangles of an n X n -5..5 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 11, 671, 401565, 2106810049, 103074513713971, 47878816111749257161
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Column 5 of A195522

Examples

			Some solutions for n=4
..0........0........0........0........0........0........0........0
..2-2......5-5......4-4......0.0......5-5.....-2.2......0.0.....-5.5
.-4.4.0...-1.3-2...-5.4.1...-1.2-1...-1.3-2....4.0-4...-1.4-3....3-5.2
..2-2.0.0.-4.2.2.0..1.0-1.0..1-2.1.0.-4.2.2.0.-2-2.4.0..1-4.3.0..2.0-2.0

A195520 Number of lower triangles of an n X n -6..6 array with all row and column sums zero.

Original entry on oeis.org

1, 1, 13, 1105, 1089411, 11131321791, 1252997799222737, 1582560262323178069903
Offset: 1

Views

Author

R. H. Hardin Sep 20 2011

Keywords

Comments

Column 6 of A195522

Examples

			Some solutions for n=4
..0........0........0........0........0........0........0........0
..0.0......0.0......0.0.....-3.3.....-5.5.....-2.2......5-5.....-5.5
..4.0-4...-4.4.0....3-2-1...-3.3.0...-1-3.4....4.0-4...-6.4.2....3-4.1
.-4.0.4.0..4-4.0.0.-3.2.1.0..6-6.0.0..6-2-4.0.-2-2.4.0..1.1-2.0..2-1-1.0
Showing 1-10 of 11 results. Next

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