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.

A339291 Number of essentially series achiral series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 13, 21, 44, 76, 158, 281, 584, 1067, 2211, 4131, 8535, 16231, 33481, 64594, 133067, 259821, 534869, 1054751, 2170736, 4316320, 8884035, 17788985, 36627593, 73776883, 151996070, 307705669, 634411061, 1289890551, 2661708319
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

See A339293 for additional details.

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(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 2: (ooooo), (o(o|oo)o).
a(6) = 4: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o).

Crossrefs

Cf. A339157, A339288 (oriented), A339290, A339292, A339293, A339294 (unoriented).

Programs

  • PARI
    \\ here B(n) gives A339290 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(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) ))); p}
seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=O(x^2)); forstep(n=2, n, 2, p=q*(1 + Z + (1 + Z)*x*Ser(EulerT(Vec(p+(s-subst(p, x, x^2))/2, 1-n))) - p)); Vec(p+O(x*x^n), -n)}

Formula

G.f.: (1 + P(x))*B(x^2) where P(x) is the g.f. of A339292 and B(x) is the g.f. of A339290.

A339292 Number of essentially parallel achiral series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 11, 21, 41, 79, 154, 304, 598, 1188, 2360, 4719, 9431, 18966, 38107, 76968, 155368, 314987, 638325, 1298379, 2640223, 5385737, 10984999, 22465570, 45945256, 94180208, 193076780, 396603802, 814838739, 1676975258, 3452212803, 7117242628
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

See A339293 for additional details.

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(3) = 1: (o|oo).
a(4) = 2: (o|ooo), (oo|oo).
a(5) = 3: (o|oooo), (oo|ooo), (o|oo|oo).
a(6) = 6: (o|ooooo), (o|o(o|oo)o), (oo|oooo), (ooo|ooo), (o|oo|ooo), (oo|oo|oo).

Crossrefs

Cf. A339158, A339289 (oriented), A339291, A339293, A339295 (unoriented).

Programs

  • PARI
    \\ here B(n) gives A339290 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(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) ))); p}
seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=Z+O(x^2), t=0); forstep(n=2, n, 2, t=q*(1 + p); p=Z + (1 + Z)*x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2, -n-1))) - t); Vec(p+O(x*x^n))}

A339296 Number of unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 60, 170, 496, 1505, 4665, 14772, 47415, 154093, 505394, 1671009, 5561274, 18615687, 62624016, 211605003, 717823582, 2443711716, 8345934897, 28587110560, 98181058020, 338029066457, 1166451261583, 4033596172701, 13975467586797, 48509872173875
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

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

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) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 9: (ooooo), (oo(o|oo)), (o(o|oo)o), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)), (oo|ooo), (o|oo|oo).

Crossrefs

Cf. A339225, A339290 (oriented), A339293 (achiral), A339294, A339295.

Programs

  • PARI
    \\ here B(n) gives A339290 as a power series.
\\ Note replacing Z by x/(1-x) gives A339225.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(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) ))); p}
seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=Z+O(x^2), t=0); forstep(n=2, n, 2, t=q*(1 + p); p=Z + (1 + Z)*x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2, -n-1))) - t); Vec(p+t+B(n,Z))/2}

Formula

a(n) = A339294(n) + A339295(n) for n > 1.
a(n) = (A339290(n) + A339293(n)) / 2.
Showing 1-4 of 4 results.

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