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 1-3 of 3 results.

A000238 Number of oriented trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, 492180, 2266502, 10598452, 50235931, 240872654, 1166732814, 5702001435, 28088787314, 139354922608, 695808554300, 3494390057212, 17641695461662, 89495023510876, 456009893224285, 2332997330210440
Offset: 1

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Author

N. J. A. Sloane

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, r(x).
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000060, A000151, A051437 (linear oriented), A334827 (oriented star-like).
Diagonal of A335362.

Programs

  • Maple
    A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(2* add(A(n-1)(x^k)/k, k=1..n-1)), x=0,n), polynom), x) fi end: a:= n-> coeff(series(A(n+1)(x) *(1-A(n+1)(x)), x=0, n+1), x,n): seq(a(n), n=1..26); # Alois P. Heinz, Aug 20 2008
  • Mathematica
    A[n_][y_] := A[n][y] = If[n == 0, 0, Normal[Series[x*Exp[2*Sum[A[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n}] /. x -> y]]; a[n_] := SeriesCoefficient[A[n+1][x]*(1-A[n+1][x]), {x, 0, n}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 12 2014, translated from Maple *)
  • PARI
    seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); Vec(Ser(A)-x*Ser(A)^2)} \\ Andrew Howroyd, May 13 2018

Formula

G.f. = x+x^2+3*x^3+8*x^4+27*x^5+... = R(x)-R(x)^2, where R(x) = g.f. for A000151.
a(n) ~ c * d^n / n^(5/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.22571615379282714232305... . - Vaclav Kotesovec, Dec 08 2014

Extensions

2 errors corrected by Paul Zimmermann, Mar 01 1996
More terms from N. J. A. Sloane, Mar 10 2007

A053873 Numbers n such that OEIS sequence A_n contains n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 14, 16, 19, 26, 27, 36, 37, 52, 59, 62, 69, 72, 115, 119, 120, 121, 134, 161, 164, 174, 177, 188, 189, 190, 193, 194, 195, 196, 209, 224, 265, 267, 277
Offset: 1

Views

Author

Jens Voß, Mar 30 2000

Keywords

Comments

A number n is in this sequence iff n appears anywhere in the terms of A_n, not just in the terms that are visible in the entry.
Is 53873 in this sequence? (A rhetorical question!) - Tanya Khovanova, Aug 09 2007
Is 53169 in this sequence? (A rhetorical question!). - Raymond Wang, Oct 07 2008
I skipped 241 since it appears that A000241(14) > 241, but as the 13th and further terms are not known this is not certain. The next term in the sequence is almost surely 319, but finding the least k for which A000319(k) = 319 requires calculating a chaotic sequence to high precision. - Charles R Greathouse IV, Jul 20 2007
241 is not in this sequence, since A000241(13) <= 225 and A000241(14) >= 0.8594*315 (see comments in A000241). - Danny Rorabaugh, Mar 13 2015

Examples

			4 is not in A000004, so 4 is not in this sequence.
60 is not in A000060, so 60 is not in this sequence.
86 is not in A000086, so 86 is not in this sequence.

Links

Crossrefs

Complement of A053169.

Extensions

More terms from N. J. A. Sloane, Aug 24 2006
a(23)-a(25) from Charles R Greathouse IV, Aug 30 2006
a(26)-a(40) from Charles R Greathouse IV, Jul 20 2007
Typo in one entry corrected by Olaf Voß, Feb 25 2008

A302939 Number of signed trees with n nodes and p positive edges. Triangle T(n,p) read by rows, 0<=p

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 3, 6, 9, 6, 3, 6, 16, 27, 27, 16, 6, 11, 37, 79, 96, 79, 37, 11, 23, 96, 233, 349, 349, 233, 96, 23, 47, 239, 679, 1187, 1439, 1187, 679, 239, 47, 106, 622, 1987, 4017, 5639, 5639, 4017, 1987, 622, 106, 235, 1607, 5784, 13216, 21263, 24758, 21263, 13216, 5784, 1607, 235
Offset: 1

Views

Author

R. J. Mathar, Apr 16 2018

Keywords

Examples

			T(2,0)=T(2,1)=1: the tree on 2 nodes (one edge) has one variant with no positive edge and one variant with one positive edge.
T(4,1)=3: the 2 trees on 4 nodes (three edges) have two variants from the linear tree with a positive edge (edge in the middle or at the end) and one variant from the star graph with one positive edge.
T(5,0)=3: there are 3 trees on 5 nodes (4 edges) where all edges are negative.
The triangle starts
    1;
    1,   1;
    1,   1,   1;
    2,   3,   3,    2;
    3,   6,   9,    6,    3;
    6,  16,  27,   27,   16,    6;
   11,  37,  79,   96,   79,   37,  11;
   23,  96, 233,  349,  349,  233,  96,  23;
   47, 239, 679, 1187, 1439, 1187, 679, 239, 47;
  106, 622,...

Links

Crossrefs

Cf. A000060 (row sums), A000055 (diagonal and 1st column), A027852 (subdiagonal and 2nd column), A304489 (rooted), A331113 (central coefficients).

Programs

  • PARI
    R(n, y)={my(v=vector(n)); v[1]=1; for(k=1, n-1, my(p=(1+y)*v[k]); my(q=Vec(prod(j=0, poldegree(p, y), (1/(1-x*y^j) + O(x*x^(n\k)))^polcoeff(p, j)))); v=vector(n, j, v[j] + sum(i=1, (j-1)\k, v[j-i*k] * q[i+1]))); v; }
M(n)={my(B=x*Ser(R(n, y))); B - (1+y)*(B^2 - substvec(B, [x, y], [x^2, y^2]))/2}
{ my(A=Vec(M(10))); for(n=1, #A, print(Vecrev(A[n]))) } \\ Andrew Howroyd, May 13 2018

Formula

T(n,p) = T(n,n-p-1), flipping all edge signs.

Extensions

Completed row 10. - R. J. Mathar, Apr 29 2018
Terms a(58) and beyond from Andrew Howroyd, May 13 2018
Showing 1-3 of 3 results.

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

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