A002026 -id:A002026 - cp's OEIS frontend

A124790 A generalized Motzkin triangle.

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 3, 4, 3, 2, 1, 0, 6, 9, 6, 5, 2, 1, 0, 15, 21, 15, 12, 6, 3, 1, 0, 36, 51, 36, 30, 15, 9, 3, 1, 0, 91, 127, 91, 76, 40, 25, 10, 4, 1, 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1

Offset: 0

Paul Barry, Nov 07 2006

Columns include A005043, A001006, A002026. Row sums are A124791. For even k, column k has g.f. x^k*M(x)^(k/2), where M(x)=2/(1-x+sqrt(1-2x-3x^2)) is the g.f. of A001006. For odd k, column k has g.f. x^k*S(x)*M(x)^floor(k/2), S(x)=(1+x-sqrt(1-2x-3x^2))/(2x(1+x)), the g.f. of A005043.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 1, 2, 1, 1,
0, 3, 4, 3, 2, 1,
0, 6, 9, 6, 5, 2, 1,
0, 15, 21, 15, 12, 6, 3, 1,
0, 36, 51, 36, 30, 15, 9, 3, 1,
0, 91, 127, 91, 76, 40, 25, 10, 4, 1,
0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
Production matrix begins
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1
- _Paul Barry_, Apr 07 2011

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Triangle is the product of A124788 and A124305, that is, it is the product of the expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2) and the inverse of the Riordan array (1,x(1-x^2)).

A366646 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^4.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 10, 20, 39, 88, 228, 600, 1507, 3652, 8866, 22100, 56365, 144656, 369784, 942480, 2408934, 6196280, 16026652, 41571640, 107959654, 280708560, 731349400, 1910098320, 4999759830, 13109582376, 34421585844, 90500370760, 238272324682

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A127902.
Cf. A213336, A364410, A366645.
Cf. A002026, A361932.

a(n) = sum(k=0, n\4, binomial(n-1, n-4*k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k) * binomial(4*k,k) / (3*k+1).

A114581 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k UDH's starting at level 0 (U=(1,1),H=(1,0),D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 2, 16, 5, 40, 10, 1, 100, 24, 3, 256, 58, 9, 663, 149, 22, 1, 1741, 386, 57, 4, 4620, 1017, 147, 14, 12376, 2702, 392, 40, 1, 33416, 7248, 1053, 113, 5, 90853, 19590, 2859, 312, 20, 248515, 53318, 7803, 870, 65, 1, 683429, 145984, 21420, 2428

Offset: 0

Emeric Deutsch, Dec 09 2005

Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 0 yields A114582. Sum(k*T(n,k),k=0..floor(n/3))=A002026(n-2).

			T(7,2)=3 because we have (UDH)(UDH)H, H(UDH)(UDH) and (UDH)H(UDH), where U=(1,1),H=(1,0),D=(1,-1) (the UDH's starting at level 0 are shown between parentheses).
Triangle starts:
1;
1;
2;
3,1;
7,2;
16,5;
40,10,1;

Cf. A001006, A114582, A002026.

G:=2/(1-z-2*t*z^3+2*z^3+sqrt(1-2*z-3*z^2)): Gser:=simplify(series(G,z=0,21)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form

G.f.=2/[1-z-2tz^3+2z^3+sqrt(1-2z-3z^2)].

A349186 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^3 * A(x)).

Original entry on oeis.org

1, 1, 3, 8, 21, 57, 157, 438, 1237, 3530, 10165, 29505, 86243, 253654, 750157, 2229469, 6655369, 19946979, 60000443, 181076982, 548125929, 1663786344, 5063133335, 15444046031, 47211447131, 144614092732, 443803262627, 1364370846941, 4201333752921, 12957168021207

Offset: 0

Ilya Gutkovskiy, Nov 09 2021

nonn

Cf. A002026, A004148, A086581, A217333, A349185.

nmax = 29; A[] = 0; Do[A[x] = (1 - x)/(1 - 2 x - x^2 - x^3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 29; CoefficientList[Series[(1 - 2 x - x^2 - Sqrt[1 - 4 x + 2 x^2 + 5 x^4])/(2 x^3), {x, 0, nmax}], x]
a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 29}]

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

a(0) = a(1) = 1; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3).

A356611 Number of SAWs spanning a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

2, 50, 2256, 292006, 124394172, 182189852062, 937116505296162, 17167376550995687961, 1130911800993488803731078, 269650395624478266477331223678, 233772496350603982679550385266064014, 739330863241806743025423160490836132227125, 8551000409049037000098287028025432585191736309022

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D4.

Cf. A001006, A002026, A007764, A116485.

A356612 Number of SAPs crossing a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

1, 3, 48, 3126, 775842, 727870836, 2575728525240, 34244061451559094, 1703999058661009145746, 316543880488539946466963896, 219157996022284922702859434801868, 564858713948847373563461482383973674774, 5415142061627863782256892670635702203299498106

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D5.

Cf. A001006, A002026, A007764, A116485.

A356613 Number of SAWs crossing a triangular domain of the hexagonal lattice.

Original entry on oeis.org

2, 7, 44, 515, 11500, 493704, 40751496, 6463642330, 1970190022696, 1154437344815284, 1300686960810345198, 2818300749120970598426, 11745284697899678209887246, 94153940687296424300453605522, 1451915619132744566900848537333082, 43072062058620235613855525243039798546

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..27
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D6.

Cf. A001006, A002026, A007764, A116485.

A356614 Number of SAWs crossing a triangular domain of the hexagonal lattice and including the top vertex.

Original entry on oeis.org

1, 3, 18, 210, 4716, 203130, 16781528, 2661898722, 811337884328, 475395297020430, 535618774376758222, 1160567857061063474508, 4836675324919658534327348, 38772333263059858336182467950, 597894854584620490267288203881970, 17736956492510173648327596231133813426

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D7.

Cf. A001006, A002026, A007764, A116485.

A356615 Number of SAPs crossing a triangular domain of the hexagonal lattice.

Original entry on oeis.org

1, 2, 9, 85, 1605, 59896, 4392639, 629739138, 175745776816, 95207239875508, 99934927799315359, 202993550188918062298, 797200289814680588454420, 6048794511036987586252009778, 88623124229469033988344357343229, 2506168305598107863294101582119745559

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D8.

Cf. A001006, A002026, A007764, A116485.

A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 3, 4, 7, 14, 27, 48, 84, 152, 284, 532, 987, 1826, 3401, 6384, 12024, 22656, 42728, 80780, 153151, 290970, 553601, 1054688, 2012373, 3845646, 7359345, 14100692, 27048061, 51941850, 99855389, 192163904, 370159216, 713672568, 1377168108, 2659729380

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A023426.

Cf. A002026, A006319, A052702.

A361229 := proc(n)
    add(binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1),k=0..floor(n/4)) ;
end proc:
seq(A361229(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

a(n) = sum(k=0, n\4, binomial(n-2*k-1, n-4*k)*binomial(2*k, k)/(k+1));

G.f.: A(x) = 2*(1-x) / (1-x+sqrt((1-x)^2-4*x^4)).
a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1).
D-finite with recurrence (n+4)*a(n) +(-3*n-7)*a(n-1) +(3*n+2)*a(n-2) +(-n+1)*a(n-3) +4*(-n+2)*a(n-4) +4*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023

cp's OEIS Frontend

A124790 A generalized Motzkin triangle.

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A366646 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^4.

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A114581 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k UDH's starting at level 0 (U=(1,1),H=(1,0),D=(1,-1)).

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A349186 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^3 * A(x)).

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Mathematica

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A356611 Number of SAWs spanning a rhomboidal domain of the hexagonal lattice.

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A356612 Number of SAPs crossing a rhomboidal domain of the hexagonal lattice.

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A356613 Number of SAWs crossing a triangular domain of the hexagonal lattice.

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A356614 Number of SAWs crossing a triangular domain of the hexagonal lattice and including the top vertex.

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A356615 Number of SAPs crossing a triangular domain of the hexagonal lattice.

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A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

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Maple

PARI

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