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 1-10 of 13 results. Next

A003430 Number of unlabeled series-parallel posets (i.e., generated by unions and sums) with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 15, 48, 167, 602, 2256, 8660, 33958, 135292, 546422, 2231462, 9199869, 38237213, 160047496, 674034147, 2854137769, 12144094756, 51895919734, 222634125803, 958474338539, 4139623680861, 17931324678301, 77880642231286, 339093495674090, 1479789701661116
Offset: 0

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Author

Keywords

Comments

Number of oriented series-parallel networks with n elements. A series configuration is a unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is a unit element or a multiset of two or more series configurations. a(n) is the number of series or parallel configurations with n elements. The sequences A007453 and A007454 enumerate respectively series and parallel configurations. - Andrew Howroyd, Dec 01 2020

Examples

			From _Andrew Howroyd_, Nov 26 2020: (Start)
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) = 5: (ooo), (o(o|o)), ((o|o)o), (o|o|o), (o|oo).
a(4) = 15: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)oo), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)), ((o|oo)o), ((o|o|o)o), (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)), (o|(o|o)o).
(End)
		

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39 (which deals with the labeled case of the same sequence).

Crossrefs

Row sums of A339231.
Column k=1 of A339228.
Cf. A000084, A003431, A048172 (labeled N-free posets), A007453, A007454, A339156, A339159, A339225.

Programs

  • Mathematica
    terms = 25; A[] = 1; Do[A[x] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms + 1}]] + O[x]^(terms + 1) // Normal, terms + 1];
    CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jun 29 2011, updated Jan 12 2018 *)
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
    seq(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x, 1-n)))); Vec(p)} \\ Andrew Howroyd, Nov 27 2020

Formula

G.f. A(x) = 1 + x + 2*x^2 + 5*x^3 + ... satisfies A(x) = exp(Sum_{k>=1} (1/k)*(A(x^k) + 1/A(x^k) - 2 + x^k)).
From: Andrew Howroyd, Nov 26 2020: (Start)
a(n) = A007453(n) + A007454(n) for n > 1.
Euler transform of A007453.
G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A007454.
(End)

Extensions

Name corrected by Salah Uddin Mohammad, Jun 07 2020
a(0)=1 prepended (using the g.f.) by Alois P. Heinz, Dec 01 2020

A339282 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks with n colored elements using exactly k colors.

Original entry on oeis.org

1, 2, 2, 4, 14, 10, 11, 84, 168, 98, 30, 522, 2109, 3004, 1396, 98, 3426, 24397, 63094, 67660, 25652, 328, 23404, 274626, 1142420, 2119985, 1805082, 576010, 1193, 165417, 3065376, 19230320, 54916745, 78809079, 55503392, 15282038, 4459, 1197934, 34201068, 311157620, 1283360335, 2761083930, 3220245007, 1932118328, 467747416
Offset: 1

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Author

Andrew Howroyd, Nov 30 2020

Keywords

Comments

Unoriented version of A339228. Equivalence is up to reversal of all parts combined in series.

Examples

			Triangle begins:
    1;
    2,     2;
    4,    14,     10;
   11,    84,    168,      98;
   30,   522,   2109,    3004,    1396;
   98,  3426,  24397,   63094,   67660,   25652;
  328, 23404, 274626, 1142420, 2119985, 1805082, 576010;
  ...
		

Crossrefs

Columns 1..2 are A339225, A339281.
Row sums are A339283.

Programs

  • PARI
    \\ R(n, k) gives colorings using at most k colors as a vector.
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
    B(n, Z)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); p}
    R(n,k)={my(Z=k*x, q=subst(B((n+1)\2, Z), x, x^2), s=subst(Z,x,x^2)+q^2/(1+q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(1 + p); p=Z + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+t-Z+B(n,Z))/2}
    M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
    {my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}

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.

A339224 Number of essentially parallel unoriented series-parallel networks with n elements.

Original entry on oeis.org

1, 1, 2, 5, 13, 41, 132, 470, 1730, 6649, 26122, 104814, 426257, 1754055, 7282630, 30470129, 128304158, 543303752, 2311904374, 9880776407, 42394198909, 182537610058, 788473887942, 3415782381520, 14837307126498, 64608442956047, 281975101347994, 1233237605651194
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: (o|o|o), (o|oo).
a(4) = 5: (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)).
		

Crossrefs

Cf. A003430, A007454 (oriented), A339158 (achiral), A339223, 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, my(t=x + q*(1 + p)); p=x + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+subst(x/(1+x), x, B(n)))/2}

Formula

a(n) = (A007454(n) + A339158(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

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.

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.

A339280 Number of unoriented series-parallel networks with n elements of 2 colors.

Original entry on oeis.org

2, 6, 22, 106, 582, 3622, 24060, 167803, 1206852, 8881775, 66496238, 504729590, 3874281594, 30020527274, 234498941338, 1844550287865, 14597849688004, 116151844649673, 928633009522942, 7456338969251761, 60101579662366508, 486145542528043029, 3944844113529346468
Offset: 1

Views

Author

Andrew Howroyd, Nov 30 2020

Keywords

Comments

See A339282 for additional details.

Examples

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 2: (1), (2).
a(2) = 6: (11), (12), (22), (1|1), (1|2), (2|2).
		

Crossrefs

Programs

  • PARI
    \\ See A339282 for R(n,k).
    seq(n) = {R(n,2)}

A339283 Number of unoriented series-parallel networks with n integer valued elements spanning an initial interval of positive integers.

Original entry on oeis.org

1, 4, 28, 361, 7061, 184327, 5941855, 226973560, 10011116097, 500553647373, 27975194135702, 1728193768303770, 116934429186262096, 8600448307248025405, 683181845460624644202, 58290243136791332908001, 5316517137637684870655592, 516199318599186277653647746
Offset: 1

Views

Author

Andrew Howroyd, Nov 30 2020

Keywords

Comments

See A339282 for additional details.

Examples

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 1: (1).
a(2) = 4: (11), (12), (1|1), (1|2).
		

Crossrefs

Row sums of A339282.

Programs

  • PARI
    \\ See A339282 for R(n,k).
    seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}

A339281 Number of unoriented series-parallel networks with n colored elements using exactly 2 colors.

Original entry on oeis.org

0, 2, 14, 84, 522, 3426, 23404, 165417, 1197934, 8847201, 66359672, 504180138, 3872043674, 30011312118, 234460670790, 1844390161675, 14597175479270, 116148990100435, 928620864502940, 7456287071153017, 60101357023288316, 486144584042042269, 3944839973878931780
Offset: 1

Views

Author

Andrew Howroyd, Nov 30 2020

Keywords

Comments

See A339282 for additional details.

Examples

			In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
a(2) = 2: (12), (1|2).
a(3) = 14: (112), (121), (122), (212), (1(1|2)), (1(2|2)), (2(1|1)), (2(1|2)), (1|12), (1|22), (2|11), (2|12), (1|1|2), (1|2|2).
		

Crossrefs

Column k=2 of A339282.

Programs

  • PARI
    \\ See A339282 for R(n,k).
    seq(n) = {R(n,2) - 2*R(n,1)}

Formula

a(n) = A339280(n) - 2*A339225(n).

A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 6, 14, 8, 1, 1, 9, 34, 39, 14, 1, 1, 12, 68, 132, 94, 20, 1, 1, 16, 126, 370, 447, 202, 30, 1, 1, 20, 212, 887, 1625, 1275, 398, 40, 1, 1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1, 1, 30, 515, 3765, 13133, 22608, 19245, 7649, 1266, 70, 1
Offset: 1

Views

Author

Andrew Howroyd, Nov 30 2020

Keywords

Comments

Unoriented version of A339231. Equivalence is up to reversal of all parts combined in series.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   5,    1;
  1,  6,  14,    8,    1;
  1,  9,  34,   39,   14,    1;
  1, 12,  68,  132,   94,   20,    1;
  1, 16, 126,  370,  447,  202,   30,   1;
  1, 20, 212,  887, 1625, 1275,  398,  40,  1;
  1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1;
  ...
T(4,0) = 1: (o|o|o|o).
T(4,1) = 4: ((o|o)(o|o)), (o(o|o|o)), (o|o|oo), (o|o(o|o)).
T(4,2) = 5: (oo(o|o)), (o(o|o)o),  (o(o|oo)),  (oo|oo), (o|ooo).
T(4,3) = 1: (oooo).
		

Crossrefs

Row sums are A339225.

Programs

  • PARI
    EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, [v^i|v<-vars])/i ))-1)}
    SubPwr(p,e)={my(vars=variables(p)); substvec(p, vars, [v^e|v<-vars])}
    BW(n, Z, W)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1+W*p)+Z)))); p}
    VertexWeighted(n, Z, W)={my(q=SubPwr(BW((n+1)\2, Z, W), 2), W2=SubPwr(W, 2), s=SubPwr(Z, 2)+W2*q^2/(1+W2*q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(W + W2*p); p=Z + x*Ser(EulerMT(Vec(t+(s-SubPwr(t, 2))/2))) - t); Vec(p+t-Z+BW(n, Z, W))/2}
    T(n)={[Vecrev(p)|p<-VertexWeighted(n, x, y)]}
    { my(A=T(12)); for(n=1, #A, print(A[n])) }

Formula

T(n,0) = T(n,n-1) = 1.
T(n,1) = A002620(n).
A339286(n) = Sum_{k=1..n-1} k*T(n,k).
Showing 1-10 of 13 results. Next