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 1-3 of 3 results.

A290586 Number of irredundant sets in the n X n rook graph.

Original entry on oeis.org

2, 11, 94, 1185, 20106, 453271, 13169346, 476777153, 20869990066, 1076251513071, 64077661097418, 4337014196039377, 329768528011095642, 27905789218764082151, 2608140451597365915346, 267506385903592339178241, 29943760423790270319833826
Offset: 1

Views

Author

Eric W. Weisstein, Aug 07 2017

Keywords

Links

Crossrefs

Main diagonal of A290818.
Row sums of A290823.
Cf. A008299, A248744.

Programs

  • Mathematica
    s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}];
c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}];
p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}];
Table[2*n^n - n! + p[n, n, 1], {n, 30}]
(* Indranil Ghosh, Aug 12 2017, after PARI code *)
  • PARI
    \\ here s(n,k) is A008299, 2*n^n - n! is A248744.
s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i);
p(m,n,x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m,k) * binomial(n,r) * k! * s(r,k) * x^r * c(m-k,n-r,x) ))}
a(n) = 2*n^n - n! + p(n,n,1); \\ Andrew Howroyd, Aug 11 2017

Formula

a(n) = 2*n^n - n! + Sum_{k=0..n-1} Sum_{r=2*k..n-1} binomial(n,k) * binomial(n,r) * k! * A008299(r,k) * c(n-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)). - Andrew Howroyd, Aug 11 2017

Extensions

a(4) corrected and a(5) from Andrew Howroyd, Aug 07 2017
Terms a(6) and beyond from Andrew Howroyd, Aug 11 2017

A290707 a(n) = 2^(n+1) + n^2 - 1.

Original entry on oeis.org

1, 4, 11, 24, 47, 88, 163, 304, 575, 1104, 2147, 4216, 8335, 16552, 32963, 65760, 131327, 262432, 524611, 1048936, 2097551, 4194744, 8389091, 16777744, 33555007, 67109488, 134218403, 268436184, 536871695, 1073742664, 2147484547, 4294968256, 8589935615
Offset: 0

Views

Author

Eric W. Weisstein, Aug 09 2017

Keywords

Comments

For n > 1, also the number of irredundant sets in the complete bipartite graph K_{n,n}.
For n > 1, also the number of irredundant sets in the 2 X n rook graph. - Andrew Howroyd, Aug 11 2017

Links

Crossrefs

Cf. A290709, A290818.

Programs

  • Mathematica
    Table[2^(n + 1) + n^2 - 1, {n, 0, 40}]
LinearRecurrence[{5, -9, 7, -2}, {4, 11, 24, 47}, {0, 20}]
CoefficientList[Series[(1 - x - 2 x^3)/((-1 + x)^3 (-1 + 2 x)), {x, 0, 20}], x]
  • PARI
    a(n)=2^(n+1)+n^2-1 \\ Charles R Greathouse IV, Aug 09 2017

Formula

a(n) = 2^(n+1) + n^2 - 1.
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).
G.f.: x*(1 - x - 2*x^3)/((-1 + x)^3*(-1 + 2*x)).
E.g.f.: exp(x)*(x^2 + x - 1 + 2*exp(x)). - Elmo R. Oliveira, Mar 06 2025

A291543 Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 11, 11, 4, 5, 18, 48, 18, 5, 6, 27, 109, 109, 27, 6, 7, 38, 218, 632, 218, 38, 7, 8, 51, 405, 1649, 1649, 405, 51, 8, 9, 66, 724, 4192, 10130, 4192, 724, 66, 9, 10, 83, 1277, 10889, 34801, 34801, 10889, 1277, 83, 10
Offset: 1

Views

Author

Andrew Howroyd, Aug 25 2017

Keywords

Comments

Maximal irredundant sets can be either dominating or not. The dominating maximal irredundant sets are the minimal dominating sets (A290632). The non-dominating maximal irredundant sets are those irredundant sets that have exactly one empty row and one empty column and at least one row and one column with more than one element. See note in A290586 for some additional details.

Examples

			Array begins:
=========================================================
m\n| 1  2    3     4      5       6        7         8
---|-----------------------------------------------------
1  | 1  2    3     4      5       6        7         8...
2  | 2  6   11    18     27      38       51        66...
3  | 3 11   48   109    218     405      724      1277...
4  | 4 18  109   632   1649    4192    10889     29480...
5  | 5 27  218  1649  10130   34801   116772    402673...
6  | 6 38  405  4192  34801  194292   856225   3804880...
7  | 7 51  724 10889 116772  856225  4730810  24810465...
8  | 8 66 1277 29480 402673 3804880 24810465 145114944...
...

Links

Crossrefs

Main diagonal is A291104.
Cf. A008299, A290586, A290632, A290818.

Programs

  • Mathematica
    T32[n_, k_] := n^k + k^n - Min[n, k]!*StirlingS2[Max[n, k], Min[n, k]];
T99[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*((k - i - j)^(n - i)/(j!*(k - i - j)!)), {j, 0, k - i}], {i, 0, k}];
T[m_, n_] /; n >= m := T32[m, n] + Sum[Sum[Binomial[m, k]*Binomial[n, j]*k!*T99[n - j, k - 1]*j!*StirlingS2[m - k, j - 1], {j, 2, m - k}], {k, 2, m - 2}]; T[m_, n_] /; n < m := T[n, m];
Table[T[m - n + 1, n], {m, 1, 10}, {n, 1, m}] // Flatten(* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
  • PARI
    \\ here s(n,k) is A008299(n,k) and b(m,n,1) is A290632(m,n).
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
b(m, n, x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
T(m, n) = b(m, n, 1) + p(m, n, 1);

Formula

T(m,n) = A290632(m, n) + Sum_{k=2..m-2} Sum_{j=2..m-k} binomial(m,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(m-k,j-1).
Showing 1-3 of 3 results.

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