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 1-4 of 4 results.

A360851 Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 6, 27, 27, 6, 10, 64, 126, 64, 10, 15, 125, 426, 426, 125, 15, 21, 216, 1125, 2208, 1125, 216, 21, 28, 343, 2493, 8830, 8830, 2493, 343, 28, 36, 512, 4872, 27456, 55700, 27456, 4872, 512, 36, 45, 729, 8676, 70434, 265635, 265635, 70434, 8676, 729, 45
Offset: 1

Views

Author

Andrew Howroyd, Feb 24 2023

Keywords

Comments

Paths of length zero are not counted here.

Examples

			Array begins:
===================================================
m\n|  1   2    3     4      5        6        7 ...
---+-----------------------------------------------
1  |  0   1    3     6     10       15       21 ...
2  |  1   8   27    64    125      216      343 ...
3  |  3  27  126   426   1125     2493     4872 ...
4  |  6  64  426  2208   8830    27456    70434 ...
5  | 10 125 1125  8830  55700   265635   961975 ...
6  | 15 216 2493 27456 265635  2006280 11158161 ...
7  | 21 343 4872 70434 961975 11158161 98309778 ...
  ...

Links

Crossrefs

Main diagonal is A360852.
Rows 1..2 are A000217(n-1), A000578.
Cf. A003991, A360199, A360850.

Programs

  • PARI
    T(m,n) = sum(j=1, min(m,n), j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j)) - m*n

Formula

T(m,n) = A360850(m,n) - A003991(m,n).
T(m,n) = -m*n + Sum_{j=1..min(m,n)} j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j).
T(m,n) = T(n,m).

A288035 Number of (undirected) paths in the complete bipartite graph K_n,n.

Original entry on oeis.org

1, 12, 135, 2224, 55725, 2006316, 98309827, 6291829440, 509638185369, 50963818537900, 6166622043087231, 887993574204562992, 150070914040571147845, 29413899151951944980364, 6618127309189187620585275, 1694240591152432030869834496, 489635530843052856921382174257
Offset: 1

Views

Author

Eric W. Weisstein, Jun 04 2017

Keywords

Links

Crossrefs

Main diagonal of A307027 and A360850.
Cf. A070968, A048617, A010790.

Programs

  • Mathematica
    Table[Sum[(n!)^2/((n - Ceiling[k/2])! (n - Floor[k/2])!), {k, 2, 2 n}], {n, 20}] (* Eric W. Weisstein, Jun 13 2017 *)
Table[n!^2 (BesselI[0, 2] + BesselI[1, 2] - HypergeometricPFQRegularized[{1}, {1 + n, 1 + n}, 1]) - n HypergeometricPFQ[{1}, {n, 1 + n}, 1], {n, 20}] // FunctionExpand (* Eric W. Weisstein, Jun 13 2017 *)
  • PARI
    a(n) = sum(k=2, 2*n, n!^2/((n-(k+1)\2)!*(n-k\2)!)); \\ Andrew Howroyd, Jun 10 2017
  • PARI
    a(n) = n!^2*sum(k=0, n-1, (1 + k)/(k!)^2) \\ Andrew Howroyd, Feb 24 2023

Formula

a(n) = Sum_{k=2..2*n} n!^2/((n-ceiling(k/2))!*(n-floor(k/2))!). - Andrew Howroyd, Jun 10 2017
a(n) = n!^2 * Sum_{k=0..n-1} (1 + k)/(k!^2). - Andrew Howroyd, Feb 24 2023

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jun 10 2017

A360873 Array read by antidiagonals: T(m,n) is the number of (non-null) connected induced subgraphs in the rook graph K_m X K_n.

Original entry on oeis.org

1, 3, 3, 7, 13, 7, 15, 51, 51, 15, 31, 205, 397, 205, 31, 63, 843, 3303, 3303, 843, 63, 127, 3493, 27877, 55933, 27877, 3493, 127, 255, 14451, 233751, 943095, 943095, 233751, 14451, 255, 511, 59485, 1938517, 15678925, 31450861, 15678925, 1938517, 59485, 511
Offset: 1

Views

Author

Andrew Howroyd, Feb 24 2023

Keywords

Examples

			Array begins:
=======================================================
m\n|  1    2      3        4          5           6 ...
---+---------------------------------------------------
1  |  1    3      7       15         31          63 ...
2  |  3   13     51      205        843        3493 ...
3  |  7   51    397     3303      27877      233751 ...
4  | 15  205   3303    55933     943095    15678925 ...
5  | 31  843  27877   943095   31450861  1033355223 ...
6  | 63 3493 233751 15678925 1033355223 67253507293 ...
  ...

Links

Crossrefs

Main diagonal is A286189.
Rows 1..2 are A000225, A360874.
Cf. A360850, A360851, A360853, A360875.

Programs

  • PARI
    \\ S is A183109, T is A262307, U is this sequence.
G(M,N=M)={ my(S=matrix(M, N), T=matrix(M, N), U=matrix(M, N));
for(m=1, M, for(n=1, N,
  S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
  T[m, n]=S[m, n]-sum(i=1, m-1, sum(j=1, n-1, T[i, j]*S[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j)));
  U[m, n]=sum(i=1, m, sum(j=1, n, binomial(m, i)*binomial(n, j)*T[i, j])) )); U
}
{ my(A=G(7)); for(n=1, #A~, print(A[n,])) }

Formula

T(m,n) = Sum_{i=1..m} Sum_{j=1..n} binomial(m, i) * binomial(n, j) * A262307(i, j).
T(m,n) = T(n,m).

A360849 Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 15, 6, 0, 0, 10, 42, 42, 10, 0, 0, 15, 90, 204, 90, 15, 0, 0, 21, 165, 660, 660, 165, 21, 0, 0, 28, 273, 1650, 3940, 1650, 273, 28, 0, 0, 36, 420, 3486, 15390, 15390, 3486, 420, 36, 0, 0, 45, 612, 6552, 45150, 113865, 45150, 6552, 612, 45, 0
Offset: 1

Views

Author

Andrew Howroyd, Feb 23 2023

Keywords

Comments

Also, T(m,n) is the number of chordless cycles of length >= 4 in the m X n rook graph.

Examples

			Array begins:
========================================================
m\n| 1  2   3    4      5       6        7         8 ...
---+----------------------------------------------------
1  | 0  0   0    0      0       0        0         0 ...
2  | 0  1   3    6     10      15       21        28 ...
3  | 0  3  15   42     90     165      273       420 ...
4  | 0  6  42  204    660    1650     3486      6552 ...
5  | 0 10  90  660   3940   15390    45150    109480 ...
6  | 0 15 165 1650  15390  113865   526155   1776180 ...
7  | 0 21 273 3486  45150  526155  4662231  24864588 ...
8  | 0 28 420 6552 109480 1776180 24864588 256485040 ...
  ...
Lower half of array as triangle T(n,k) for 1 <= k <= n begins:
  0;
  0,  1;
  0,  3,  15;
  0,  6,  42,  204;
  0, 10,  90,  660,  3940;
  0, 15, 165, 1650, 15390, 113865;
  0, 21, 273, 3486, 45150, 526155, 4662231;
  ...

Links

Crossrefs

Rows 1..3 are A000004, A000217(n-1), A059270(n-1).
Main diagonal is A070968.
Cf. A269562, A286418, A360850 (paths), A360853.

Programs

  • PARI
    T(m,n) = sum(j=2, min(m,n), binomial(m,j)*binomial(n,j)*j!*(j-1)!/2)

Formula

T(m,n) = Sum_{j=2..min(m,n)} binomial(m,j)*binomial(n,j)*j!*(j-1)!/2.
T(m,n) = T(n,m).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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