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.

A280786 Number of topologically distinct sets of n circles with one pair intersecting.

Original entry on oeis.org

1, 4, 15, 50, 162, 506, 1558, 4727, 14227, 42521, 126506, 374969, 1108476, 3269902, 9630631, 28328999, 83251569, 244471484, 717486860, 2104777227, 6172357873, 18096097750, 53044095421, 155464365080, 455601800970, 1335107222743, 3912330438784, 11464463809180, 33595343643160
Offset: 2

Views

Author

N. J. A. Sloane, Jan 20 2017

Keywords

Links

Crossrefs

Row sums of A280787.
Column k=1 of A261070.

Programs

  • Maple
    A280786 := proc(N)
    if N < 2 then
        0;
    else
        add(A280787(N,f),f=1..N-1) ;
    end if;
end proc:
A280787 := proc(N,f)
    option remember ;
    local Npr,ct ;
    if f = N then
        return 0;
    elif f = N-1 then
        return 1;
    elif f = 1 then
        A280786(N-1)+A280788(N-2) ;
    else
        ct := 0 ;
        for Npr from 1 to N-1 do
            ct := ct+procname(Npr,1)*A033185(N-Npr,f-1) ;
        end do:
        ct ;
    end if;
end proc:
seq(A280786(n),n=2..30) ; # R. J. Mathar, Mar 06 2017
  • Mathematica
    a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np]*a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
A280788[n_] := If[n == 0, 1, Sum[a81[np + 1]*A027852[n - np + 2], {np, 0, n}]];
t[n_] := t[n] = Module[{d, j}, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n - j], {j, 1, n - 1}]/(n - 1)]];
b[1, 1, 1] = 1;
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[t[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]]; A033185[n_, k_] := b[n, n, k];
A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
A280787[n_, f_] := A280787[n, f] = Module[{ct}, Which[f == n, Return[0], f == n - 1, Return[1], f == 1, Return[A280786[n - 1] + A280788[n - 2]], True, ct = 0; Do[ct += A280787[np, 1]*A033185[n - np, f - 1], {np, 1, n - 1}]]; ct];
Table[A280786[n], {n, 2, 30}] (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar and Alois P. Heinz *)

A339303 Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 9, 6, 6, 2, 1, 20, 16, 15, 8, 3, 1, 48, 37, 41, 22, 12, 3, 1, 115, 96, 106, 69, 38, 15, 4, 1, 286, 239, 284, 194, 124, 52, 20, 4, 1, 719, 622, 750, 564, 377, 189, 77, 24, 5, 1, 1842, 1607, 2010, 1584, 1144, 618, 292, 100, 30, 5, 1
Offset: 1

Views

Author

Andrew Howroyd, Dec 04 2020

Keywords

Comments

Linear forests (A339067) are considered up to reversal of the linear order.
T(n,k) is the number of unlabeled trees on n nodes rooted at two indistinguishable nodes at distance k-1 from each other.

Examples

			Triangle read by rows:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   2,   1;
    9,   6,   6,   2,   1;
   20,  16,  15,   8,   3,   1;
   48,  37,  41,  22,  12,   3,  1;
  115,  96, 106,  69,  38,  15,  4,  1;
  286, 239, 284, 194, 124,  52, 20,  4, 1;
  719, 622, 750, 564, 377, 189, 77, 24, 5, 1;
  ...

Links

Crossrefs

Columns 1..4 are A000081, A027852, A280788(n-3), A339302.
Row sums are A303840(n+2).
Row sums excluding the first column are A303833.
Cf. A339067.

Programs

  • PARI
    \\ TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(r^k + r^(k%2)*subst(r, x, x^2)^(k\2), -n)/2}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

Formula

G.f of column k: (r(x)^k + r(x)^(k mod 2)*r(x^2)^floor(k/2))/2 where r(x) is the g.f. of A000081.

A280787 Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.

Original entry on oeis.org

1, 3, 1, 10, 4, 1, 30, 15, 4, 1, 91, 50, 16, 4, 1, 268, 162, 55, 16, 4, 1, 790, 506, 185, 56, 16, 4, 1, 2308, 1558, 594, 190, 56, 16, 4, 1, 6737, 4727, 1878, 617, 191, 56, 16, 4, 1, 19609, 14227, 5825, 1970, 622, 191, 56, 16, 4, 1
Offset: 2

Views

Author

N. J. A. Sloane, Jan 20 2017

Keywords

Examples

			Triangle begins:
     1;
     3,    1;
    10,    4,   1;
    30,   15,   4,   1;
    91,   50,  16,   4,  1;
   268,  162,  55,  16,  4,  1;
   790,  506, 185,  56, 16,  4, 1;
  2308, 1558, 594, 190, 56, 16, 4, 1;
...

Links

Crossrefs

Row sums give A280786.

Programs

  • Mathematica
    a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np]*a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
t[n_] := t[n] = Module[{d, j}, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n - j], {j, 1, n - 1}]/(n - 1)]];
b[1, 1, 1] = 1;
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[t[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]]; A033185[n_, k_] := b[n, n, k];
A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
A280787[n_, f_] := A280787[n, f] = Module[{ct}, Which[f == n, Return[0], f == n - 1, Return[1], f == 1, Return[A280786[n - 1] + A280788[n - 2]], True, ct = 0; Do[ct += A280787[np, 1]*A033185[n - np, f - 1], {np, 1, n - 1}]]; ct];
Table[A280787[n, f], {n, 2, 11}, {f, 1, n - 1}] // Flatten (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar and Alois P. Heinz *)
Showing 1-3 of 3 results.

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

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