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Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.
10

%I #20 Jul 30 2021 08:34:08

%S 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,2,2,

%T 2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,8,11,8,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,2,11,42,74,76,38,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,74,296,633,768,558,219,50

%N Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

%H Sean A. Irvine, <a href="/A049337/b049337.txt">Table of n, a(n) for n = 1..469; rows 1..14 flattened</a>

%e Triangle begins

%e 0;

%e 0,0;

%e 0,0,0,0;

%e 0,0,0,1,0,0,0;

%e 0,0,0,0,0,1,1,0,0,0;

%e 0,0,0,0,0,0,1,2,2,...;

%e ...

%e From _Hugo Pfoertner_, Nov 24 2020: (Start)

%e Transposed table:

%e .

%e Nodes Sums

%e 4 5 6 7 8 9 10 11 12 13 14 |A002840

%e Edges-+--+--+---+---+----+-----+------+------+-----+---+|-------

%e 6 | 1 . . . . . . . . . . | 1

%e 7 | . . . . . . . . . . . | 0

%e 8 | . 1 . . . . . . . . . | 1

%e 9 | . 1 1 . . . . . . . . | 2

%e 10 | . . 2 . . . . . . . . | 2

%e 11 | . . 2 2 . . . . . . . | 4

%e 12 | . . 2 8 2 . . . . . . | 12

%e 13 | . . . 11 11 . . . . . . | 22

%e 14 | . . . 8 42 8 . . . . . | 58

%e 15 | . . . 5 74 74 5 . . . . | 158

%e 16 | . . . . 76 296 76 . . . . | 448

%e 17 | . . . . 38 633 633 38 . . . | 1342

%e 18 | . . . . 14 768 2635 768 14 . . | 4199

%e 19 | . . . . . 538 6134 6134 558 . . | 13384

%e 20 | . . . . . 219 8822 25626 8822 219 . | 43708

%e 21 | . . . . . 50 7916 64439 64439 7916 50 | 144810

%e .. | . . . . . . .. .. .. .. .. | ..

%e ---+--+--+---+---+----+-----+------+-------+----+---+

%e Sums 1 2 7 34 257 2606 32300 440564 6384634 .. A000944

%e (End)

%Y Cf. A000944, A002840, A021103, A003094, A049334.

%Y A049337, A058787, A212438 are all versions of the same triangle.

%Y Cf. A058788.

%K nonn,tabf,nice

%O 1,33

%A _Brendan McKay_

%E Missing zeros inserted by _Sean A. Irvine_, Jul 29 2021