A339290 -id:A339290 - cp's OEIS frontend

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

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

Andrew Howroyd, Dec 07 2020

nonn

See A339290 for additional details.

			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).

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

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)}

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.

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

Andrew Howroyd, Dec 07 2020

nonn

See A339290 for additional details.

			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).

Cf. A339155, A339288, A339290, A339292 (achiral), A339295 (unoriented).

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))}

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

A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874

Offset: 0

N. J. A. Sloane, Dec 20 2000

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. See A000084 for examples.

Order is not considered significant in series configurations. - Andrew Howroyd, Dec 22 2020

			From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) 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) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
Index entries for sequences mentioned in Moon (1987)

A000084 is the case that multiple edges are allowed.
A058381 is the case that edges are labeled.
A339290 is the case that order is significant in series configurations.
Cf. A058385, A058386, A000311, A000669, A006351.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020

a(n) = A058385(n) + A058386(n).

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

Andrew Howroyd, Dec 07 2020

nonn

See A339293 for additional details.

			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).

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

\\ 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)}

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

Andrew Howroyd, Dec 07 2020

nonn

See A339293 for additional details.

			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).

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

\\ 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

Andrew Howroyd, Dec 07 2020

nonn

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.

			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).

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

\\ 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))}

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

Andrew Howroyd, Dec 07 2020

nonn

See A339296 for additional details.

			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)).

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

\\ 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}

a(n) = (A339288(n) + A339291(n)) / 2.

A339295 Number of essentially parallel unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 1, 2, 4, 10, 25, 69, 197, 589, 1806, 5685, 18168, 58905, 192904, 637294, 2119994, 7094961, 23865782, 80642017, 273571625, 931389949, 3181184007, 10897272983, 37429033777, 128874546753, 444744161951, 1538030244174, 5329246656885, 18499283612755

Offset: 1

Andrew Howroyd, Dec 07 2020

nonn

See A339296 for additional details.

			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) = 4: (o|oooo), (o|o(o|oo)), (oo|ooo), (o|oo|oo).

Cf. A339224, A339289 (oriented), A339292 (achiral), A339294, A339296.

\\ 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+1-1/(1+B(n,Z)))/2}

a(n) = (A339289(n) + A339292(n)) / 2.

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

Andrew Howroyd, Dec 07 2020

nonn

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.

			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).

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

\\ 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}

a(n) = A339294(n) + A339295(n) for n > 1.

a(n) = (A339290(n) + A339293(n)) / 2.

A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 2, 12, 108, 1380, 22440, 446040, 10461360, 282970800, 8670594240, 296850597120, 11230473925440, 465262142304960, 20948652798353280, 1018583225567107200, 53190962586022060800, 2969038807022050963200, 176410305542414738995200, 11116489894884127122969600

Offset: 1

Andrew Howroyd, Dec 22 2020

nonn

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 labeled 1..n.

			a(3) = 12 because there are 2 unlabeled structures each of which can be labeled in 6 ways. The unlabeled structures are (ooo) and (o|oo).

A048172 is the case with multiple unit elements in parallel allowed.
A058381 is the case that order is not significant in series configurations.
Main diagonal of A339297.
Cf. A339290 (unlabeled), A339299, A339300.

\\ Note giving Z=exp(x)-1 gives A048172.
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))}

seq(n)={my(A=O(x*x^n)); Vec(serlaplace(subst(serreverse(log(1+x+A) - x^2/(1+x)), x, log(1+x+A))))}

a(n) = A339299(n) + A339300(n).
E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.
E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.

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

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

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A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

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

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

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

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

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A339295 Number of essentially parallel unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

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