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 1-4 of 4 results.

A380886 Triangle T(n,k), 1<=k<=n: column k are the coefficients of the INVERT transform of Sum_{i=1..k} i*x^i.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 11, 17, 21, 1, 21, 42, 50, 55, 1, 43, 100, 128, 138, 144, 1, 85, 235, 323, 358, 370, 377, 1, 171, 561, 813, 923, 965, 979, 987, 1, 341, 1331, 2043, 2378, 2510, 2559, 2575, 2584, 1, 683, 3158, 5150, 6125, 6527, 6681, 6737, 6755, 6765, 1, 1365, 7503, 12967, 15772, 16972, 17441, 17617, 17680, 17700, 17711
Offset: 1

Views

Author

R. J. Mathar, Feb 07 2025

Keywords

Examples

			The full array starts
  1    1    1    1    1    1    1    1    1    1
  1    3    3    3    3    3    3    3    3    3
  1    5    8    8    8    8    8    8    8    8
  1   11   17   21   21   21   21   21   21   21
  1   21   42   50   55   55   55   55   55   55
  1   43  100  128  138  144  144  144  144  144
  1   85  235  323  358  370  377  377  377  377
  1  171  561  813  923  965  979  987  987  987
  1  341 1331 2043 2378 2510 2559 2575 2584 2584
  1  683 3158 5150 6125 6527 6681 6737 6755 6765
but the non-interesting upper right triangular part is not put into the sequence.

Crossrefs

Cf. A001045 (column k=2), A101822 (column k=3), A322059 (column k=4?), A001906 (diagonal), A054452 (subdiagonal).

Programs

  • Maple
    A380886 := proc(n,k)
    local g,x ;
    g := 1/(1-add(i*x^i,i=1..k)) ;
    coeftayl(g,x=0,n) ;
end proc:
seq(seq( A380886(n,k),k=1..n),n=1..12) ;

Formula

T(n,k) = [x^n] 1/(1-x^1-2*x^2-3*x^3-4*x^4-...-k*x^k) .

A380890 Triangle T(n,k) read by rows: the number of graphs with n nodes which are enriched cycles (necklaces) and the elements in the cycle are marked linear chains up to length k.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 5, 7, 1, 3, 7, 9, 12, 1, 5, 14, 18, 21, 24, 1, 5, 19, 29, 35, 38, 42, 1, 8, 35, 56, 71, 77, 81, 85, 1, 10, 60, 100, 133, 148, 156, 160, 165, 1, 15, 107, 192, 264, 297, 317, 325, 330, 335, 1, 19, 187, 361, 511, 586, 630, 650, 660, 665, 671, 1, 31, 352, 714, 1041, 1206, 1306, 1350, 1375, 1385, 1391, 1397
Offset: 1

Views

Author

R. J. Mathar, Feb 07 2025

Keywords

Comments

The number of marked/rooted linear chains (linear trees) with up to k nodes has the generating function x+ x^2 +2*x^3 +2*x^4 + 3*x^5 +3*x^6+...+floor((k+1)/2)*x^k, a truncated version of A110654. [The node in the chain that is marked can be at the end of the chain, penultimate node etc up to the middle node, which gives floor((k+1)/2) ways; marking any of the other nodes does not generate more marked chains because the entire chain can be flipped over.] A multiset of such marked chains defines an "enriched" necklace if m marked nodes are arranged in a circle and a chord is wired through these nodes. That chord is the only cycle in the graph. The total number n of nodes in that graph is the sum over the individual chain lengths of the m chains.
In Bower's nomenclatures the columns approach the CIK transform of A110654 for k->oo: 1, 2, 4, 7, 12, 24, 42, 85, 165, 335, 671, 1397,...
If the total number of nodes n is fixed, the number of graphs T(n,k) stays constant in the rows at k>n, because chains longer than n are never used due to the limit imposed on n.
The book of Bona (edt.) shows on page 1002 a graph with a cycle constructed with 5 linear chains up to length k=4 (1 chain of length 1, 2 chains of length 2, 1 chain of length 3, 1 chain of length 4) and n=12 nodes in total. It seems the "pointed operation" there creates more chains than enumeration of rooted linear trees here.

Examples

			The full array starts
   1    1    1    1    1    1    1    1    1    1    1    1
   1    2    2    2    2    2    2    2    2    2    2    2
   1    2    4    4    4    4    4    4    4    4    4    4
   1    3    5    7    7    7    7    7    7    7    7    7
   1    3    7    9   12   12   12   12   12   12   12   12
   1    5   14   18   21   24   24   24   24   24   24   24
   1    5   19   29   35   38   42   42   42   42   42   42
   1    8   35   56   71   77   81   85   85   85   85   85
   1   10   60  100  133  148  156  160  165  165  165  165
   1   15  107  192  264  297  317  325  330  335  335  335
   1   19  187  361  511  586  630  650  660  665  671  671
   1   31  352  714 1041 1206 1306 1350 1375 1385 1391 1397
The sequence gathers only the non-trivial lower left triangle up to the diagonal.
T(1,k)=1 means whatever the set of variable-length linear chains is, if the total number of nodes is 1 only the linear chain of length 1 (ring of a single element) is a necklace.
T(n,1)=1 means the only available element in the necklace are simple (marked) nodes. To generate a enriched necklace with n nodes in total can only be done by putting that node n times repeated on a circle.
T(4,3)=5 means the available chains up to length k=3 are x, x-o, x-o-o and o-x-o; x denotes the marked node. The 5 cycles/necklaces with n=4 nodes are: [x]-[x]-[x]-[x], [x]-[x]-[x-o], [x-o]-[x-o], [x]-[x-o-o], and [x]-[o-x-o], where [...] is the a linear node in the circle.

Links

Crossrefs

Cf. A322059, A000358 (column k=2, binary necklace).

Programs

  • Maple
    # Generating function of a marked linear chain
# (=rooted linear tree) up to n nodes.
# Essentially a truncated (polynomial) version of A110654.
# @param n Number of nodes
# @param x free variable in the o.g.f.
gLinChain := proc(n,x)
    add(floor((i+1)/2) *x^i,i=1..n) ;
end proc:
# k: max element in each chain
# n: number of total nodes in graph
A380890 := proc(n,k)
    local x,g ,cidx,d,gd,ncyc;
    # number of possible elements in the cycle
    g := gLinChain(k,x) ;
    # wrap g into the Polya cycle index of the cyclic group C_ncyc
    cidx := 0 ;
    for ncyc from 1 to n do
        for d in numtheory[divisors](ncyc) do
            gd := subs(x=x^d,g) ;
            cidx := cidx+1/ncyc*numtheory[phi](d)*gd^(ncyc/d) ;
        end do:
    end do:
    coeff(cidx,x^n) ;
end proc:
seq(seq(A380890(n,k),k=1..n),n=1..12) ;

A322060 Expansion of generating function related to a certain class of combinatorial objects.

Original entry on oeis.org

1, 4, 10, 26, 54, 144
Offset: 1

Views

Author

N. J. A. Sloane, Dec 25 2018

Keywords

Comments

For precise definition see Example 15.3.6 of Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 1001-1002.

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 1001, 1002.

Links

Crossrefs

Cf. A322059.

A322752 Number of "funny trees" on n nodes.

Original entry on oeis.org

0, 2, 3, 9, 30, 110, 423, 1706, 7085, 30186, 131071, 578194, 2583377, 11667874, 53180604, 244301512, 1129947243, 5257592237, 24592945975, 115578827200, 545478791124, 2584216074295, 12285025045259, 58584860422121, 280181867792399, 1343499543045511, 6457845289959966
Offset: 0

Views

Author

N. J. A. Sloane, Dec 25 2018

Keywords

Comments

For precise definition see Example 15.3.7 of Bona (2015).
The trees considered here have nodes of two types: black and white. The child nodes of black nodes are unordered and can be either black or white. The child nodes of white nodes are linearly ordered and must be black. - Andrew Howroyd, Feb 06 2025

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 1002.

Links

Crossrefs

Cf. A277996.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(w=O(x),b=O(x)); for(n=1, n, w=x/(1-b); b=x*(1 + x*Ser(EulerT(Vec(b+w))))); Vec(b+w, -n-1)} \\ Andrew Howroyd, Feb 06 2025

Extensions

Offset corrected and a(7) onwards from Andrew Howroyd, Feb 06 2025
Showing 1-4 of 4 results.

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

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