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.

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.

A339289 Number of essentially parallel oriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 1, 2, 5, 14, 39, 117, 353, 1099, 3458, 11066, 35738, 116622, 383448, 1269869, 4230557, 14170956, 47693457, 161207066, 546987882, 1862464911, 6361729689, 21793247587, 74855427331, 257743707769, 889477338903, 3076038022778, 10658447368514, 36998473045302
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(1) = 1: (o).
a(3) = 1: (o|oo).
a(4) = 2: (o|ooo), (oo|oo).
a(5) = 5: (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).

Crossrefs

Cf. A339155, A339288, A339290, A339292 (achiral), A339295 (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(1-1/(1+p))}

Formula

G.f.: B(x)/(1 + B(x)) where B(x) is the g.f. of A339290.

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.

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.
Showing 1-5 of 5 results.

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