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.

A108247 Duplicate of A108243.

Original entry on oeis.org

1, 1, 10, 760, 190050, 103050570, 102359800620, 168076482974400
Offset: 0

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Keywords

A333351 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n labeled nodes, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 0, 6, 0, 1, 1, 0, 1, 1, 10, 22, 15, 1, 1, 0, 1, 0, 15, 0, 130, 0, 1, 1, 0, 1, 1, 21, 158, 760, 822, 105, 1, 1, 0, 1, 0, 28, 0, 3355, 0, 6202, 0, 1, 1, 0, 1, 1, 36, 654, 12043, 93708, 190050, 52552, 945, 1, 1, 0, 1, 0, 45, 0, 36935, 0, 3535448, 0, 499194, 0, 1
Offset: 0

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Author

Andrew Howroyd, Mar 15 2020

Keywords

Examples

			Array begins:
=================================================================
n\k | 0   1    2      3       4        5         6          7
----+------------------------------------------------------------
  0 | 1   1    1      1       1        1         1          1 ...
  1 | 1   0    0      0       0        0         0          0 ...
  2 | 1   1    1      1       1        1         1          1 ...
  3 | 1   0    1      0       1        0         1          0 ...
  4 | 1   3    6     10      15       21        28         36 ...
  5 | 1   0   22      0     158        0       654          0 ...
  6 | 1  15  130    760    3355    12043     36935     100135 ...
  7 | 1   0  822      0   93708        0   3226107          0 ...
  8 | 1 105 6202 190050 3535448 45163496 431400774 3270643750 ...
  ...

Links

Crossrefs

Rows n=4..6 are A000217(n+1), A244868 (with interspersed zeros), A244878.
Columns k=0..4 are A000012, A123023, A002137, A108243 (with interspersed zeros), A367497.
Cf. A059441 (graphs), A333157, A333330 (unlabeled nodes), A333467 (with loops).

Programs

  • PARI
    MultigraphsByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r, h, p, q, v, e) = if(!p, if(ok(x^e+q, r), acc(x^e+q, v)), my(i=poldegree(p), t=pollead(p)); self()(r, limit, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(k=1, min(t, (limit-e)\m), self()(r, if(k==t, limit, i+m-1), p-k*x^i, q+k*x^(i+m), binomial(t, k)*v, e+k*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, limit, src[i, 1], 0, src[i, 2], 0))); Mat(M);
}
T(n,k)={if((n%2&&k%2)||(n==1&&k>0), 0, vecsum(MultigraphsByDegreeSeq(n, k, (p,r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[,2]))}
{ for(n=0, 8, for(k=0, 7, print1(T(n,k), ", ")); print) }

A110039 Number of 3-regular labeled graphs on 2n vertices with no multiple edges, but loops are allowed. (3-regular = trivalent and a loop incident on a vertex counts as two edges.)

Original entry on oeis.org

1, 1, 8, 730, 188790, 102737670, 102172297920, 167870491048260, 423971126389110300, 1559445481095305703900, 8010574937878696134151200, 55572909620219147733302926200, 506607333530572584517841616582600, 5931728848766374810152582924943605000
Offset: 0

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Author

Marni Mishna, Jul 08 2005

Keywords

Comments

Also the same as n X n symmetric matrices with {0,2}-entries on the diagonal and entries from {0,1} elsewhere, with row sum equal to 3.

Examples

			a(1)=1: {(1,1), (1,2), (2,2)}

References

  • Goulden, I. P.; Jackson, D. M. Labelled graphs with small vertex degrees and $P$-recursiveness. SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093)

Crossrefs

Cf. A108246, A001147, A002829, A108243, A005814.

Programs

  • Mathematica
    max = 30; f[x_] := Sum[a[n]*(x^n/n!), {n, 0, max}]; a[0] = 1; a[1] = 1; coef = CoefficientList[ 9*x^3*(x^4 - 2)*f''[x] + 3*(x^10 - 2*x^8 - 5*x^6 - 18*x^2 + 8)*f'[x] - x*(x^4 - 4*x^2 + 2)*(x^6 - 2*x^2 + 12)*f[x], x]; Table[a[n], {n, 0, max, 2}]/. Solve[Thread[coef[[2 ;; max]] == 0]][[1]] (* Vaclav Kotesovec, Sep 15 2014 *)

Formula

Differential equation satisfied by the e.g.f. F(t) = sum_n a(n)/2n! t^n: {F(0) = 1, (-t^5+4*t^4+52*t-20*t^2-24)*F(t) + (-144*t+48-12*t^3-12*t^4+6*t^5)*(d/dt)F(t) + (36*t^4-72*t^2)*(d^2/dt^2)F(t)}.
Recurrence: {(123200*n^9 + 30135960*n + 8448*n^10 + 256*n^11 + 105258076*n^3 + 4989600 + 53358140*n^5 + 75458988*n^2 + 91991460*n^4 + 21100464*n^6 + 5718768*n^7 + 1045440*n^8)*a(n) + (-24948000 - 12736*n^9 - 90804600*n - 384*n^10 - 134879084*n^3 - 32082204*n^5 - 145393020*n^2 - 80308236*n^4 - 8713656*n^6 - 1589856*n^7 - 186624*n^8)*a(n + 1) + (11840760*n + 6932520*n^3 + 4989600 + 544320*n^5 + 12084468*n^2 + 2446668*n^4 + 74592*n^6 + 5760*n^7 + 192*n^8)*a(n + 2) + (-1108800 - 2428000*n - 1014166*n^3 - 44740*n^5 - 2148828*n^2 - 278430*n^4 - 3912*n^6 - 144*n^7)*a(n + 3) + (-6435 - 3887*n - 780*n^2 - 52*n^3)*a(n + 4) + (3003 + 1635*n + 297*n^2 + 18*n^3)*a(n + 5) - 3*a(n + 6)}.
Goulden and Jackson give a differential equation satisfied by the e.g.f, which presumably agrees with the above. - N. J. A. Sloane, Sep 02 2013
Recurrence (for n > 5): 3*(9*n^2 - 27*n + 16)*a(n) = 3*(2*n - 1)*(27*n^4 - 108*n^3 + 138*n^2 - 63*n + 4)*a(n-1) - (n-1)*(2*n - 3)*(2*n - 1)*(3*n - 4)*(18*n^2 - 27*n - 13)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(27*n^3 - 90*n^2 + 57*n + 8)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(9*n^2 - 9*n - 2)*a(n-4). - Vaclav Kotesovec, Sep 15 2014
a(n) ~ sqrt(2) * 6^n * n^(3*n) / exp(3*n). - Vaclav Kotesovec, Sep 15 2014

Extensions

More terms from Vaclav Kotesovec, Sep 15 2014

A108285 Triangle read by rows, generated from (1, 2, 3, ...).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 11, 10, 1, 6, 18, 26, 15, 1, 7, 27, 58, 57, 21, 1, 8, 38, 112, 179, 120, 28, 1, 9, 51, 194, 453, 543, 247, 36
Offset: 0

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Author

Gary W. Adamson, May 30 2005

Keywords

Comments

By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.
Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

Examples

			4th column (offset) = 10, 26, 58, 112, ...= f(x), x = 1, 2, 3; x^3 + 2x^2 + 3x + 4.
First few rows of the triangle are:
  1;
  1, 3;
  1, 4, 6;
  1, 5, 11, 10;
  1, 6, 18, 26, 15;
  1, 7, 27, 58, 57, 21;
  1, 8, 38, 112, 179, 120, 28;
  ...

Crossrefs

Cf. A108286, A000295, A000349, A014825, A000217, A108283, A108284, A108286.

Formula

n-th column = f(x), x = 1, 2, 3, ...; x^(n) + 2*x^(n-1) + 3*x^(n-2) + ... + (n+1).

A109542 a(n) = number of labeled 3-regular (trivalent) multi-graphs without self-loops on 2n vertices with a maximum of 2 edges between any pair of nodes. Also a(n) = number of labeled symmetric 2n X 2n matrices with {0,1,2}-entries with row sum equal to 3 for each row and trace 0.

Original entry on oeis.org

0, 7, 640, 170555, 94949400, 95830621425, 159062872168200, 404720953797785625
Offset: 1

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Author

Jeremy Gardiner, Aug 29 2005

Keywords

Examples

			a(2)=7 because for 2*n=4 nodes there are 7 possible labeled graphs whose adjacency matrices are as follows:
0 2 1 0
2 0 0 1
1 0 0 2
0 1 2 0;
0 1 2 0
1 0 0 2
2 0 0 1
0 2 1 0;
0 2 0 1
2 0 1 0
0 1 0 2
1 0 2 0;
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0;
0 0 2 1
0 0 1 2
2 1 0 0
1 2 0 0;
0 1 0 2
1 0 2 0
0 2 0 1
2 0 1 0;
0 0 1 2
0 0 2 1
1 2 0 0
2 1 0 0.

Crossrefs

Cf. A001205, A002829, A108243.

Extensions

a(5)-a(8) from Max Alekseyev, Aug 30 2005
Showing 1-5 of 5 results.

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

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