A378341 -id:A378341 - cp's OEIS frontend

A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 3, 3, 4, 3, 0, 0, 2, 4, 7, 8, 7, 10, 8, 0, 0, 0, 4, 8, 15, 19, 22, 19, 29, 23, 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68, 0, 0, 0, 0, 8, 25, 47, 82, 104, 150, 175, 200, 176, 266, 215, 0, 0, 0, 0, 7, 26, 64, 123, 186, 288, 346, 488, 556, 634, 557, 844, 680

Offset: 3

Andrew Howroyd, Nov 25 2024

See A378103 for illustration of initial terms. This sequence counts only those maps which have mirror symmetry.
The planar maps considered are without loops or isthmuses.
The number of edges is n + k - 1.

			Triangle begins:
n\k | 1  2  3  4   5   6   7   8   9  10  11  12  13
----+------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1, 2,  1;
  6 | 0, 0, 2, 3,  3,  4,  3;
  7 | 0, 0, 2, 4,  7,  8,  7, 10,  8;
  8 | 0, 0, 0, 4,  8, 15, 19, 22, 19, 29, 23;
  9 | 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
Andrew Howroyd, PARI Program, Nov 2024.

Row sums are A378339.
Column sums are A378341.
Antidiagonal sums are A378342.
Cf. A378103 (unsensed), A378336 (sensed), A002712.

my(A=A378340rows(10)); for(i=1, #A, print(A[i])) \\ See Links for program.

T(n,2*n-5) = A002712(n-3). - Ya-Ping Lu, Dec 16 2024

A378339 Number of n-node connected achiral planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 15, 46, 139, 449, 1448, 4799, 15971, 54107, 184048, 633365, 2188644, 7624153, 26659774, 93772014, 330946542, 1173233805, 4171572088, 14884905583, 53249784377, 191043396323, 686956509123, 2476044740337, 8942288625630, 32360306616473, 117309644377341

Offset: 3

Andrew Howroyd, Nov 24 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..200

Row sums of A378340.

Cf. A377785 (unsensed), A378335 (sensed), A378341 (by faces), A378342 (by edges).

A378339seq(30) \\ See PARI link in A378340 for program code.

A378342 Number of n edge connected achiral planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 6, 7, 15, 22, 33, 59, 94, 140, 249, 388, 617, 1040, 1675, 2628, 4481, 7108, 11494, 19030, 30885, 49315, 82559, 132406, 215326, 353923, 576552, 928437, 1539504, 2485102, 4051828, 6636368, 10830526, 17524893, 28903185, 46838466, 76464837, 125019717

Offset: 3

Andrew Howroyd, Nov 25 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100

Antidiagonal sums of A378340.

Cf. A378338 (sensed), A378339, A378341, A378343 (unsensed).

A378338seq(45) \\ See PARI link in A378340 for program code.

A378337 Number of connected sensed planar maps with an external face and n internal triangular faces.

Original entry on oeis.org

1, 2, 6, 32, 155, 1043, 7083, 51983, 393064, 3069809, 24522342, 199837021, 1655355627, 13907476161, 118275808160, 1016657106858, 8821284816805, 77181267608093, 680346219098844, 6037537131435871, 53904120217352519, 483921982054548241, 4366269495618350828, 39577142154691122910

Offset: 1

Andrew Howroyd, Nov 23 2024

nonn

Column sums of A378336.

Cf. A378335, A378338, A378341.

A378337seq(20) \\ See PARI link in A378340 for program code.

A378190 Number of planar maps with an external face and n internal triangular faces.

Original entry on oeis.org

1, 2, 6, 24, 100, 586, 3725, 26532, 198081, 1539550, 12274565, 99959181, 827795678, 6954099320, 59138955508, 508331799502, 4410651891166, 38590663253312, 340173195849485, 3018768835038348, 26952060900042852, 241960993507098580, 2183134755112963493, 19788571100313277286

Offset: 1

Ya-Ping Lu, Nov 19 2024

nonn

Column sums of A378103.

Cf. A002713, A169808, A377785.

a(n) = (A378337(n) + A378341(n))/2.

cp's OEIS Frontend

A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

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A378339 Number of n-node connected achiral planar maps with an external face in which all internal faces are triangles.

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A378342 Number of n edge connected achiral planar maps with an external face in which all internal faces are triangles.

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A378337 Number of connected sensed planar maps with an external face and n internal triangular faces.

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A378190 Number of planar maps with an external face and n internal triangular faces.

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