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 1-5 of 5 results.

A339159 Number of achiral series-parallel networks with n elements.

Original entry on oeis.org

1, 2, 3, 7, 12, 29, 54, 130, 258, 616, 1274, 3030, 6458, 15287, 33335, 78694, 174587, 411469, 925246, 2179010, 4952389, 11662221, 26733827, 62980863, 145385388, 342766624, 795810810, 1878109984, 4381423357, 10352044123, 24247955489, 57362089607
Offset: 1

Views

Author

Andrew Howroyd, Nov 27 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. 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 of series-parallel networks, 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) = 2: (oo), (o|o).
a(3) = 3: (ooo), (o|oo), (o|o|o), (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
a(4) = 7: (oooo), ((o|o)(o|o)), (o(o|o)o).

Crossrefs

Cf. A003430 (oriented), A339157, A339158, A339225 (unoriented).

Programs

  • PARI
    \\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2), t=p); for(n=1, n\2, t=x + q*(1 + p); p=x + x*Ser(EulerT(Vec(t+(s-subst(t,x,x^2))/2))) - t); Vec(p+t-x+O(x*x^n))}

Formula

a(n) = A339157(n) + A339158(n) for n > 1.

A339158 Number of essentially parallel achiral series-parallel networks with n elements.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 37, 84, 180, 413, 902, 2084, 4628, 10726, 24128, 56085, 127421, 296955, 680092, 1588665, 3662439, 8574262, 19875081, 46628789, 108584460, 255264307, 596774173, 1405626896, 3297314994, 7780687159, 18305763571, 43271547808, 102069399803
Offset: 1

Views

Author

Andrew Howroyd, Nov 27 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. a(n) is the number of parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.

Examples

			In the following examples of series-parallel networks, 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: (o|o).
a(3) = 2: (o|oo).
a(4) = 4: (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
a(5) = 8: (o|oooo), (o|(o|o)(o|o)), (o|o(o|o)o), (oo|ooo), (o|o|ooo), (o|oo|oo), (o|o|o|oo), (o|o|o|o|o).
a(6) = 16 includes (o(o|o)|(o|o)o) which is the first example of a network that is achiral but does not have reflective symmetry when embedded in the plane as shown below (edges correspond to elements):
               A
             /   \\
            o      o   --- No reflective symmetry ---
             \\  /
               Z

Crossrefs

Cf. A003430, A007454 (oriented), A339157, A339159, A339224 (unoriented).

Programs

  • PARI
    \\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, my(t=x + q*(1 + p)); p=x + x*Ser(EulerT(Vec(t+(s-subst(t,x,x^2))/2))) - t); Vec(p+O(x*x^n))}

Formula

G.f.: x - S(x) - 1 + exp(Sum_{k>=1} (S(x^k) + (R(x^(2*k)) - S(x^(2*k)))/2)/k) where S(x) is the g.f. of A339157 and R(x) is the g.f. of A007453.

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.

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.

A339223 Number of essentially series unoriented series-parallel networks with n elements.

Original entry on oeis.org

1, 1, 2, 6, 17, 57, 196, 723, 2729, 10638, 42161, 169912, 692703, 2853523, 11852644, 49592966, 208800209, 883970867, 3760605627, 16068272965, 68925340187, 296705390322, 1281351319402, 5549911448062, 24103086681839, 104938476264310, 457920147387969, 2002462084788769
Offset: 1

Views

Author

Andrew Howroyd, Nov 27 2020

Keywords

Comments

See A339225 for additional details.

Examples

			In the following examples of series-parallel networks, 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), (o|o).
a(3) = 2: (ooo), (o(o|o)).
a(4) = 6: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)).

Crossrefs

Cf. A003430, A007453 (oriented), A339157 (achiral), A339224, A339225.

Programs

  • PARI
    \\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, p = x + q*(1 + x + x*Ser(EulerT(Vec(p+(s-subst(p, x, x^2))/2))) - p)); Vec(p+x+subst(x^2/(1+x),x,B(n)))/2}

Formula

a(n) = (A007453(n) + A339157(n))/2.
Showing 1-5 of 5 results.

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

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