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.

A339290 Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 306, 930, 2887, 9100, 29082, 93951, 306414, 1007361, 3335088, 11108986, 37203873, 125193694, 423099557, 1435427202, 4886975378, 16690971648, 57172387872, 196358421066, 676050576441, 2332887221847, 8067160995797, 27950871439353, 97019613539949
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements.

Examples

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 5: (oooo), (o(o|oo)), ((o|oo)o), (o|ooo), (oo|oo).
a(5) = 13: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o), (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).

Links

Crossrefs

A003430 is the case with multiple unit elements in parallel allowed.
A058387 is the case that order is not significant in series configurations.
Cf. A339156, A339288, A339289, A339293 (achiral), A339296 (unoriented), A339301 (labeled).

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}

Formula

a(n) = A339288(n) + A339289(n).
G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339289.

A339297 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements and without multiple unit elements in parallel using exactly k colors.

Original entry on oeis.org

1, 1, 2, 2, 12, 12, 5, 64, 162, 108, 13, 354, 1734, 2760, 1380, 36, 1992, 16977, 48716, 56100, 22440, 103, 11538, 161691, 746316, 1488240, 1338120, 446040, 306, 68427, 1524969, 10652086, 32760180, 49718640, 36614760, 10461360, 930, 414294, 14382720, 146464740, 652517010, 1487453760, 1816345440, 1131883200, 282970800
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. T(n, k) is the number of series or parallel configurations with n unit elements of k colors using each color at least once.

Examples

			Triangle begins:
    1;
    1,     2;
    2,    12,     12;
    5,    64,    162,    108;
   13,   354,   1734,   2760,    1380;
   36,  1992,  16977,  48716,   56100,   22440;
  103, 11538, 161691, 746316, 1488240, 1338120, 446040;
  ...

Crossrefs

Column 1 is A339290.
Main diagonal is A339301.
Row sums are A339298.
Cf. A339228.

Programs

  • PARI
    \\ R(n, k) gives colorings using at most k colors as a vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(Z=k*x, p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}

A339299 Number of essentially series oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

0, 2, 6, 72, 840, 14040, 276360, 6494880, 175452480, 5375311200, 183962227680, 6958070380800, 288200792880000, 12974113884251520, 630742839699772800, 32933429270386444800, 1838083950894102912000, 109201772719684867622400, 6880730833827011402841600
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

See A339301 for additional details.

Crossrefs

A058349 is the case with multiple unit elements in parallel allowed.
A058380 is the case that order is not significant in series configurations.
Cf. A339288 (unlabeled), A339300, A339301.

Programs

  • PARI
    seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(p-p/(1+p)), -n)}

Formula

E.g.f.: P(x)^2/(1 - P(x)) where P(x) is the e.g.f. of A339300.
E.g.f.: B(x)^2/(1 + B(x)) where B(x) is the e.g.f. of A339301.
E.g.f.: B(log(1+x)) where x + B(x) is the e.g.f. of A058349.

A339300 Number of essentially parallel oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 6, 36, 540, 8400, 169680, 3966480, 107518320, 3295283040, 112888369440, 4272403544640, 177061349424960, 7974538914101760, 387840385867334400, 20257533315635616000, 1130954856127948051200, 67208532822729871372800, 4235759061057115720128000
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

See A339301 for additional details.

Crossrefs

A048174 is the case with multiple edges in parallel allowed.
A058379 is the case that order is not significant in series configurations.
Cf. A339289 (unlabeled), A339299, A339301.

Programs

  • PARI
    seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(1-1/(1+p)))}

Formula

E.g.f. (1 + x)*exp(S(x)) - S(x) - 1 where S(x) is the e.g.f. of A339299.
E.g.f.: B(x)/(1 + B(x)) where B(x) is the e.g.f. of A339301.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048174.
Showing 1-4 of 4 results.

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