A055907 -id:A055907 - cp's OEIS frontend

A055915 Column 9 of triangle A055907.

1, 18, 164, 1050, 5367, 23232, 88125, 299992, 932849, 2685747, 7235237, 18392853, 44427807, 102552852, 227298680, 485679518, 1003915611, 2013360324, 3927664296, 7469755903, 13876848040, 25225619858, 44939663514, 78569415198, 134974052794, 228088973049

Offset: 9

Christian G. Bower, Jun 22 2000

nonn

A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999 (H_9 for hexagonal lattices Table 1).

G.f.: x^9*B(x)/((1-x)^19*(1+x)^9*(1+x^2)^8*(1+x^2+x^4)^6*(1+x^4)^4*(1+x^2+x^4+x^6+x^8)^2) where B(x) is g.f. of A055918.

a(31) corrected and more terms from Sean A. Irvine, Apr 09 2022

A055913 Column 7 of triangle A055907.

Original entry on oeis.org

1, 14, 100, 511, 2116, 7501, 23527, 66834, 174953, 427486, 984654, 2154809, 4508722, 9067082, 17600002, 33094234, 60466665, 107632267, 187074805, 318119850, 530176267, 867294295, 1394495598, 2206460931, 3439326144

Offset: 7

Christian G. Bower, Jun 22 2000

nonn

A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999 (H_7 for hexagonal lattices Table 1).

G.f.: x^7*B(x)/((1-x)^15*(1+x)^7*(1+x^2)^6*(1+x^2+x^4)^4*(1+x^4)^2) where B(x) is g.f. of A055916.

A055914 Column 8 of triangle A055907.

Original entry on oeis.org

1, 16, 130, 748, 3459, 13612, 47141, 147116, 421040, 1119827, 2796446, 6610818, 14893276, 32150509, 66807075, 134139807, 261101457, 494074770, 911085320, 1640685080, 2890653798, 4990940719, 8456978339, 14081784408, 23068258050

Offset: 8

Christian G. Bower, Jun 22 2000

nonn

A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999 (H_8 for hexagonal lattices Table 1).

G.f.: x^8*B(x)/((1-x)^17*(1+x)^8*(1+x^2)^7*(1+x^2+x^4)^5*(1+x^4)^3*(1+x^2+x^4+x^6+x^8)) where B(x) is g.f. of A055917.

A055908 Column 2 of triangle A055907.

Original entry on oeis.org

1, 4, 10, 21, 39, 67, 107, 163, 238, 337, 463, 622, 818, 1058, 1346, 1690, 2095, 2570, 3120, 3755, 4481, 5309, 6245, 7301, 8484, 9807, 11277, 12908, 14708, 16692, 18868, 21252, 23853, 26688, 29766, 33105, 36715, 40615, 44815, 49335, 54186

Offset: 2

Christian G. Bower, Jun 14 2000

nonn

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (H_2 for hexagonal lattices of Section 6).

Cf. A055907.

G.f.: x^2*(1 + x + x^3)/((1 - x)^5*(1+x)^2*(1 + x^2)).

A055909 Column 3 of triangle A055907.

Original entry on oeis.org

1, 6, 20, 53, 121, 249, 471, 836, 1409, 2276, 3544, 5350, 7862, 11286, 15866, 21896, 29721, 39746, 52436, 68331, 88047, 112287, 141841, 177604, 220577, 271880, 332752, 404572, 488860, 587292, 701700, 834096, 986673, 1161822, 1362132

Offset: 3

Christian G. Bower, Jun 14 2000

nonn

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (H_3 for hexagonal lattices of Section 6).

Cf. A055907.

G.f.: x^3*(1+x)*(1+x+3x^3-x^4+x^5) / ((1-x)^7*(1+x)^3*(1+x^2)^2).

A055910 Column 4 of triangle A055907.

Original entry on oeis.org

1, 8, 36, 128, 388, 1050, 2601, 6013, 13141, 27416, 55012, 106816, 201685, 371828, 671630, 1192093, 2084360, 3597991, 6143128, 10391369, 17439285, 29073358, 48199071, 79535657, 130740732, 214230558, 350125952, 571019573, 929690344

Offset: 4

Christian G. Bower, Jun 14 2000

nonn

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (H_4 for hexagonal lattices of Section 6).

Cf. A055907.

G.f.: x^4*(1+3x+4x^2+10x^3+12x^4+14x^5+16x^6+13x^7+14x^8+7x^9+6x^10+4x^11+x^13)/ ((1-x)^9*(1+x)^4*(1+x^2)^3*(1-x-x^2)*(1+x+x^2)).

A055911 Column 5 of triangle A055907.

Original entry on oeis.org

1, 10, 56, 240, 854, 2654, 7412, 18974, 45173, 101074, 214361, 433731, 841921, 1574765, 2849215, 5002411, 8546675, 14243335, 23202995, 37016099, 57925111, 89043169, 134635403, 200468817, 294253144, 426179565, 609587498, 861767228

Offset: 5

Christian G. Bower, Jun 15 2000

nonn

The given generating function does not appear to generate the sequence. - Sean A. Irvine, Apr 09 2022

A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999 (H_5 for hexagonal lattices Table 1).

Cf. A055907.

G.f.: x^5*(1 + 4x + 8x^2 + 22x^3 + 37x^4 + 56x^5 + 88x^6 + 98x^7 + 137x^8 + 118x^9 + 145x^10 + 113x^11 + 104x^12 + 79x^13 + 50x^14 + 40x^15 + 17x^16 + 10x^17 + 6x^18 + x^20) / ((1 - x)^11 * (1 + x)^5 * (1 + x^2)^4 * (1 + x^2 + x^4)^2).

A055912 Column 6 of triangle A055907.

Original entry on oeis.org

1, 12, 74, 331, 1212, 3829, 10778, 27651, 65745, 146663, 309831, 624423, 1207701, 2252494, 4067393, 7134576, 12191298, 20342987, 33217622, 53174235, 83580380, 129177701, 196557754, 294776792, 436141322, 637204826, 920020667

Offset: 6

Christian G. Bower, Jun 22 2000

nonn

A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999 (H_6 for hexagonal lattices Table 1).

Cf. A055907.

G.f.: x^6(1 + 5x + 13x^2 + 40x^3 + 85x^4 + 160x^5 + 297x^6 + 453x^7 + 711x^8 + 956x^9 + 1276x^10 + 1614x^11 + 1798x^12 + 2192x^13 + 2064x^14 + 2446x^15 + 1949x^16 + 2220x^17 + 1546x^18 + 1642x^19 + 1011x^20 + 959x^21 + 558x^22 + 427x^23 + 249x^24 + 135x^25 + 85x^26 + 32x^27 + 15x^28 + 8x^29 + x^31)/((1 - x)^13(1 + x)^6(1 + x^2)^5(1 + x^2 + x^4)^3(1 + x^4)). [See column t6,m in table 1 p. 12 of Guttmann and Conway.]

A055898 Triangle: Number of directed site animals on a square lattice with n+1 total sites and k sites supported in one particular way.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 16, 7, 1, 1, 15, 39, 31, 9, 1, 1, 21, 81, 101, 51, 11, 1, 1, 28, 150, 272, 209, 76, 13, 1, 1, 36, 256, 636, 696, 376, 106, 15, 1, 1, 45, 410, 1340, 1980, 1496, 615, 141, 17, 1, 1, 55, 625, 2600, 5000, 5032, 2850, 939, 181, 19, 1, 1

Offset: 0

Christian G. Bower, Jun 13 2000

			Triangle begins:
  1;
  1, 1;
  1, 3, 1;
  1, 6, 5, 1;
  1,10,16, 7, 1;
  ...

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (A_sq of Section 6).
M. Bousquet-Mélou, New enumerative results on two-dimensional directed animals
M. Bousquet-Mélou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.

Row sums give A005773. Columns 0-8: A000012, A000217, A011863(n-1), A055899-A055904. Cf. A055905, A055907.

nmax = 10;
A[x_, y_] = (1/2) x ((1 - (4 x/((1 + x) (1 + x - x y))))^(-1/2) - 1);
g = A[x, y] + O[x]^(nmax+3);
row[n_] := CoefficientList[Coefficient[g, x, n+2], y];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Jul 24 2018 *)

T(n,m):=sum(binomial(n-k,m)*sum(binomial(n-m-k,i)*(-1)^(n+m-i)*binomial(k+i,k)*binomial(2*i+1,i+1),i,0,n-m-k),k,0,n); /* Vladimir Kruchinin, Jan 26 2022 */

G.f.: A(x, y)=(1/2x)((1-(4x/((1+x)(1+x-xy))))^(-1/2) - 1).

T(n,m) = Sum_{k=0..n} C(n-k,m)*Sum_{i=0..n-m-k} (-1)^(n+m-i) *C(n-m-k,i) *C(k+i,k) *C(2*i+1,i+1). - Vladimir Kruchinin, Jan 26 2022

A055919 Number of directed site animals on hexagonal net (honeycomb) with n sites.

Original entry on oeis.org

2, 4, 10, 24, 60, 152, 402, 1094, 3012, 8232, 22086

Offset: 1

Christian G. Bower, Jun 22 2000

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
A. J. Guttmann and A. R. Conway, Hexagonal lattice directed site animals, Statistical Physics on the Eve of the Twenty-First Century, ed. M. T. Batchelor, World Scientific, 1999.

Row sums of A055907.

Name edited by Andrey Zabolotskiy, Jun 21 2022

cp's OEIS Frontend

A055915 Column 9 of triangle A055907.

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A055913 Column 7 of triangle A055907.

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A055914 Column 8 of triangle A055907.

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A055908 Column 2 of triangle A055907.

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A055909 Column 3 of triangle A055907.

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A055910 Column 4 of triangle A055907.

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A055911 Column 5 of triangle A055907.

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A055912 Column 6 of triangle A055907.

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A055898 Triangle: Number of directed site animals on a square lattice with n+1 total sites and k sites supported in one particular way.

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A055919 Number of directed site animals on hexagonal net (honeycomb) with n sites.

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