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.

Previous Showing 11-15 of 15 results.

A122801 Number of labeled bipartite graphs on 2n vertices having equal parts and no isolated vertices.

Original entry on oeis.org

1, 1, 21, 2650, 1452605, 3149738046, 26503552820514, 868081172737564500, 111606080497500509325405, 56762846667123360827351083510, 114847831981827229530824587617895286, 927685362544629192461621864598358779955500, 29976424929810726580224613882836823991388901138994
Offset: 0

Views

Author

Max Alekseyev, Sep 11 2006

Keywords

Links

Crossrefs

Cf. A122802, A048291, A052332, A001831, A002031, A047863, A001700.

Programs

  • PARI
    { A122801(n) = binomial(2*n-1,n) * sum(k=0, n, binomial(n, k) * (-1)^k * (2^(n-k)-1)^n ); }

Formula

For n>0, a(n) = A001700(n-1) * A048291(n) = A052332(2n) - A122802(2n).

Extensions

Terms a(11) and beyond from Andrew Howroyd, Nov 07 2019

A122802 Number of labeled bipartite graphs on n vertices with no isolated vertices.

Original entry on oeis.org

1, 0, 1, 6, 29, 510, 5032, 161406, 3294405, 194342910, 7934652356, 881008805886, 71275547085536, 15178191426486270, 2434250064518832302, 1008694542117649154046, 321680912414994434144165, 262063364967549826752315390, 166681427053102507699172431372
Offset: 0

Views

Author

Max Alekseyev, Sep 11 2006

Keywords

Links

Crossrefs

Cf. A122801, A052332, A001831, A002031, A047863.

Programs

  • PARI
    a(n)={sum(k=0, n, binomial(n, k)*(2^k-2)^(n-k)) - if(n%2==0&&n>0, binomial(n-1, n/2)*sum(k=0, n/2, binomial(n/2, k)*(-1)^k*(2^(n/2-k)-1)^(n/2)))} \\ Andrew Howroyd, Nov 07 2019

Formula

a(2n+1) = A052332(2n+1); a(2n) = A052332(2n) - A122801(n) for n > 0.

Extensions

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A360718 Number of idempotent Boolean relation matrices on [n] that have no proper power primitive.

Original entry on oeis.org

1, 2, 9, 52, 459, 5526, 91161, 2039024, 62264215, 2618031658, 153147765333, 12544274587956, 1443661355799075, 233590364506712318, 53152637809972391281, 17010099259539378971368, 7660283773351147860024879, 4856904906875123474086041426
Offset: 0

Views

Author

Geoffrey Critzer, Feb 24 2023

Keywords

Comments

A Boolean relation matrix R is said to be convergent in its powers if in the sequence {R, R^2, R^3, ...} there is an m such that R^m = R^(m+1).
An idempotent Boolean relation matrix E is said to have a proper power primitive iff there is a convergent relation R with limit matrix E where R is not equal to E.
Let P = C_1 + C_2 + ... + C_k + S be a poset with rank(P) <= 1 (A001831) where each C_i is a weakly connected component of size 2 or more and S is a set of isolated points. Let A be a subset of [n] and let E = P - {(x, x): x in A}. Then E is an idempotent relation with no proper power primitive iff A satisfies exactly one of the following conditions:
i) A is a nonempty subset of domain(E) and A contains at most one point in domain(C_i) for 1 <= i <= k.
ii) A is a nonempty subset of image(E) and A contains at most one point in image(C_i) for 1 <= i <= k.
iii) A contains at most one point in S.
The first term in the e.g.f. below counts the number of such relations for which condition i) or ii) is satisfied. The second term in the e.g.f. counts the number of such relations for which condition iii) is satisfied. - Geoffrey Critzer, Feb 11 2024

Links

Crossrefs

Cf. A360743, A001831, A121337, A002031.

Programs

  • Mathematica
    nn = 17; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}]; c[x_] := Log[A[x]] - x; Range[0, nn]! CoefficientList[Series[2 (Exp[x D[c[x], x]/2] - 1) Exp[c[x]] Exp[x] + Exp[c[x]] D[x Exp[x], x], {x, 0, nn}], x]

Formula

E.g.f.: 2(exp(x * c'(x)/2) - 1) exp(c(x)) exp(x) + exp(c(x))*(x exp(x))' where c(x) is the e.g.f. for A002031.

A174122 Partial sums of A001831.

Original entry on oeis.org

1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737
Offset: 0

Views

Author

Jonathan Vos Post, Mar 08 2010

Keywords

Comments

Partial sums of number of labeled graded partially ordered sets with n elements. The subsequence of primes in this partial sum begins: 2, 5, 12589.

Crossrefs

Cf. A001831, A002031, A047863, A052332, A001833, A048194, A052296, A135753, A135754.

Formula

a(n) = SUM[i=0..n] A001831(i) = SUM[i=0..n] SUM[j=0..i] ((-1)^j*C(n,j)*A047863(j)).

A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 39, 87, 348, 24, 841, 4205, 480, 11643, 69858, 9420, 240, 227893, 1595251, 206640, 9240, 6285807, 50286456, 5389552, 299040, 3360, 243593041, 2192337369, 172041408, 9848160, 211680
Offset: 0

Views

Author

Geoffrey Critzer, Feb 11 2024

Keywords

Examples

			 Triangle begins
      1;
      1,       1;
      3,       6;
     13,      39;
     87,     348,     24;
    841,    4205,    480;
  11643,   69858,   9420,  240;
 227893, 1595251, 206640, 9240;
 ...

Links

Crossrefs

Cf. A360718 (row sums), A001831 (column k=0), A360743 (T(n,0) + T(n,1) ), A151817 (T(2n,n) for n>=2), A002031.

Programs

  • Mathematica
    nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}];
c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &,
 Range[0, nn]! CoefficientList[
   Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] +
     Exp[c[ x]] (y x Exp[  x] + Exp[ x]), {x, 0, nn}], {x, y}]]

Formula

E.g.f.: 2(exp(y*x*c'(x)/2)-1)*exp(c(x))*exp(x) + exp(c(x))*(y*x*exp(x) + exp(x)) where c(x) is the e.g.f. for A002031.
Previous Showing 11-15 of 15 results.

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