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.

A339288 Number of essentially series oriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

0, 1, 1, 3, 8, 22, 64, 189, 577, 1788, 5642, 18016, 58213, 189792, 623913, 2065219, 6878429, 23032917, 77500237, 261892491, 888439320, 3024510467, 10329241959, 35379140285, 121502993735, 418306868672, 1443409882944, 4991122973019, 17292424070839, 60021140494647, 208684858267921
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

See A339290 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) = 3: (oooo), (o(o|oo)), ((o|oo)o).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o).

Crossrefs

Cf. A339154, A339289, A339290, A339291 (achiral), A339294 (unoriented).

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 - p/(1+p), -n)}

Formula

G.f.: P(x)^2/(1 - P(x)) where P(x) is the g.f. of A339289.
G.f.: B(x)^2/(1 + B(x)) where 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))}

A339293 Number of achiral series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 1, 2, 3, 5, 10, 17, 34, 62, 123, 230, 462, 879, 1772, 3427, 6930, 13562, 27501, 54338, 110449, 219962, 448054, 898146, 1833248, 3694974, 7556473, 15301319, 31349605, 63734241, 130807801, 266853663, 548599872, 1122544408, 2311386319, 4742103354, 9778950947
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 series or parallel configurations with n unit elements that are invariant under the reversal of all contained 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) = 3: (oooo), (o|ooo), (oo|oo).
a(5) = 5: (ooooo), (o(o|oo)o), (o|oooo), (oo|ooo), (o|oo|oo).
a(6) = 10: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o), (o|ooooo), (o|o(o|oo)o), (oo|oooo), (ooo|ooo), (o|oo|ooo), (oo|oo|oo).

Crossrefs

Cf. A339159, A339290 (oriented), A339291, A339292, A339296 (unoriented).

Programs

  • PARI
    \\ here B(n) gives A339290 as a power series.
\\ Note replacing Z by x/(1-x) gives A339159.
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+O(x*x^n))}

Formula

a(n) = A339291(n) + A339292(n) for n > 1.

A339294 Number of essentially series unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

0, 1, 1, 2, 5, 13, 35, 101, 299, 916, 2859, 9087, 29247, 95188, 312490, 1033715, 3441280, 11520726, 38758234, 130962986, 444251957, 1512321767, 5164750890, 17689837577, 60752024243, 209154519704, 721707099632, 2495565928527, 8646220929912, 30010588561120
Offset: 1

Views

Author

Andrew Howroyd, Dec 07 2020

Keywords

Comments

See A339296 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) = 2: (oooo), (o(o|oo)).
a(5) = 5: (ooooo), (oo(o|oo)), (o(o|oo)o), (o(o|ooo)), (o(oo|oo)).

Crossrefs

Cf. A339157, A339288 (oriented), A339290, A339291 (achiral), A339295, A339296.

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)); my(t=B(n, Z)); Vec(p + t - t/(1+t), -n)/2}

Formula

a(n) = (A339288(n) + A339291(n)) / 2.
Showing 1-4 of 4 results.

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