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A338859 Square array A(m,k) is the number of unicyclic graphs with m trees of k nodes; m,k >= 0, read by falling antidiagonals. 1
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 9, 10, 4, 1, 1, 0, 1, 20, 45, 20, 6, 1, 1, 0, 1, 48, 210, 165, 55, 8, 1, 1, 0, 1, 115, 1176, 1540, 1035, 136, 13, 1, 1, 0, 1, 286, 6670, 19600, 22155, 6273, 430, 18, 1, 1, 0, 1, 719, 41041, 260130, 692076, 324008, 46185, 1300, 30, 1, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

The number of unicyclic graphs with m k-trees is equal to the number of bracelets with m beads using up to A000081(k) colors, so A(m,k) = A321791(m, A000081(k)).

Because A102911(k) is the number of graphs constituted by 2 k-node rooted trees with the roots joined by an edge, A(2,k) = A102911(k). [Bomfim illustration for k=2,3].

Column 1 refers to Cyclic graphs, Column 2 refers to Sunlet graphs.

LINKS

Table of n, a(n) for n=0..77.

Index entries for sequences related to bracelets

Washington Bomfim, Illustraction of graphs counted by A(2,k), k=2,3

Eric Weisstein's World of Mathematics, Sunlet graph

FORMULA

A(m,k) = A321791(m, A000081(k)).

EXAMPLE

A begins,

---+------------------------------------------------------------------------------

m/k|0 1 2  3    4      5        6           7              8                 9

---+------------------------------------------------------------------------------

0  |1 1 1  1    1      1        1           1              1                 1 ...

1  |0 1 1  2    4      9       20          48            115               286 ...

2  |0 1 1  3   10     45      210        1176           6670             41041 ...

3  |0 1 1  4   20    165     1540       19600         260130           3939936 ...

4  |0 1 1  6   55   1035    22155      692076       22247785         842202361 ...

5  |0 1 1  8  136   6273   324008    25535712     2012117671      191362445560 ...

6  |0 1 1 13  430  46185  5376070  1020580232   192799298140    45606942211831 ...

7  |0 1 1 18 1300 344925 91508580 41936107248 19000229453710 11179807512382366 ...

...|           ...            ...            ...            ...            ...

---+------------------------------------------------------------------------------

The A(3,3) = 4 unicyclic graphs with 3 trees of 3 nodes

         0                                  0

         |                                  |

         0                0   0             0             0   0

         |                 \ /              |              \ /

         0                  0               0               0

        /*\                /*\             /*\             /*\

       /***\              /***\           /***\           /***\

      0-----0            0---- 0         0-----0         0-----0

     /       \          / \   / \       / \   / \        |     |

    0         0        0   0 0   0     0   0 0   0       0     0

   /           \                                         |     |

  0             0                                        0     0

The graphs above are also representations of bracelets with m = 3 beads using up to A000081(k=3) = 2 colors.

PROG

(PARI)                              \\ From Robert A. Russell formula of A321791.

A(m, k)={ if( m == 0, return(1),

(k^((m+1)>>1)+k^ceil((m+1)/2)) / 4 + sumdiv(m, d, eulerphi(d)*k^(m/d) )/(m<<1)) };

seq(max_m) = { my(f = vector(max_m), kk, mm, ff); f[1] = 1;

for(j=1, max_m - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1]));

print1(A(0, 0) ", "); for(k = 1, max_m, kk = k; mm = 0; ff = f[kk];

until(A(mm, ff)==0, print1(A(mm, ff)", "); mm++; kk--; if(kk==0, ff=0, ff = f[kk]) );

print1("0, ")) };

CROSSREFS

Cf. A000081 (row 1), A321791, A102911 (row 2), A000029 (column 3), A032275 (column 4).

Sequence in context: A144903 A108934 A108947 * A152459 A275784 A331508

Adjacent sequences:  A338856 A338857 A338858 * A338860 A338861 A338862

KEYWORD

nonn,tabl

AUTHOR

Washington Bomfim, Nov 24 2020

STATUS

approved

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Last modified April 20 12:29 EDT 2021. Contains 343135 sequences. (Running on oeis4.)