A023431 -id:A023431 - cp's OEIS frontend

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

1, 0, 1, 2, 2, 4, 8, 13, 25, 49, 91, 177, 349, 681, 1349, 2693, 5377, 10806, 21820, 44163, 89721, 182868, 373616, 765341, 1571551, 3233690, 6667242, 13772469, 28498419, 59065838, 122606998, 254865837, 530507839, 1105663034, 2307131590, 4819623077, 10079039819, 21099213611, 44211213545

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Cf. A000245, A023426, A023431, A090345, A346503, A346504, A357308.

a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 38}]
nmax = 38; A[] = 0; Do[A[x] = 1 + x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A364539 G.f. satisfies A(x) = 1 + xA(x) + x^3A(x)^5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 62, 182, 583, 1928, 6358, 21063, 70888, 241889, 831634, 2874584, 9995579, 34966279, 122938956, 434062141, 1538378816, 5471697241, 19525345791, 69880082323, 250767909528, 902123110483, 3252793321513, 11753570922933, 42553831219830

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A063019, A182454, A349311, A364522.

Cf. A023431, A071879, A215340.

a(n) = sum(k=0, n\3, binomial(n+2*k, 5*k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,5*k) * binomial(5*k,k) / (4*k+1).

A126218 Triangle read by rows: T(n,k) is the number of 0-1-2 trees (i.e., ordered trees with all vertices of outdegree at most two) with n edges and k pairs of adjacent vertices of outdegree 2.

Original entry on oeis.org

1, 1, 2, 4, 7, 2, 13, 8, 26, 20, 5, 52, 50, 25, 104, 130, 75, 14, 212, 322, 217, 84, 438, 770, 644, 294, 42, 910, 1836, 1806, 952, 294, 1903, 4362, 4830, 3108, 1176, 132, 4009, 10268, 12738, 9576, 4188, 1056, 8494, 24032, 33219, 27948, 14760, 4752, 429, 18080

Offset: 0

Emeric Deutsch, Dec 24 2006

Row n has floor(n/2) terms (n >= 2).
Row sums are the Motzkin numbers (A001006).
T(n,1) = A023431(n+1).
Sum_{k=0..floor(n/2)-1} k*T(n,k) = 2*A014532(n-3) (n >= 4).

			Triangle starts:
   1;
   1;
   2;
   4;
   7,  2;
  13,  8;
  26, 20,  5;
  52, 50, 25;

Cf. A001006, A014532, A023431.

G:=1/2*(2*z^2*t^2-z+4*z^3*t-2*z^3*t^2-2*z^2*t-2*z^3+1-sqrt(1+4*z^3*t-4*z^2*t+z^2-2*z-4*z^3))/z^2/(z*t-t-z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: 1;1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od; # yields sequence in triangular form

G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*(1 + zG + t(G-1-zG))^2 (see the Maple program for the explicit expression).

A166284 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDD's (0<=k<=floor(n/2); U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 3, 1, 7, 7, 3, 13, 17, 6, 1, 26, 36, 16, 4, 52, 77, 45, 10, 1, 104, 173, 111, 30, 5, 212, 387, 268, 95, 15, 1, 438, 857, 666, 266, 50, 6, 910, 1911, 1641, 714, 175, 21, 1, 1903, 4287, 3975, 1940, 546, 77, 7, 4009, 9619, 9606, 5205, 1610, 294, 28, 1

Offset: 0

Emeric Deutsch, Oct 11 2009

T(n,k) is also the number of weighted lattice paths B(n) having k (1,0)-steps of weight 2. B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: a (1,0)-step of weight 1; a (1,0)-step of weight 2; a (1,1)-step of weight 2; a (1,-1)-step of weight 1. The weight of a path is the sum of the weights of its steps. Example: row 3 is 2,2; indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the four paths of weight 3 are ud, hH, Hh, and hhh, having 0, 1, 1, and 0 (1,0)-steps of weight 2, respectively. - Emeric Deutsch, Aug 23 2014
Row n contains 1+floor(n/2) entries.
Sum of entries in row n is A004148(n+1) (the secondary structure numbers).
T(n,0) = A023431(n).
Sum(k*T(n,k), k=0..floor(n/2)) = A110320(n-1).

			T(5,2)=3 because we have UDUUDDUUDD, UUDDUDUUDD, and UUDDUUDDUD.
Triangle starts:
1;
1;
1,1;
2,2;
4,3,1;
7,7,3;
13,17,6,1;
26,36,16,4;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. - _Emeric Deutsch_, Aug 23 2014

Cf. A004148, A023431, A110320.

F := RootOf(G = 1+z*G+t*z^2*G+z^3*G^2, G): Fser := series(F, z = 0, 18): for n from 0 to 15 do P[n] := sort(coeff(Fser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f. G=G(t,z) satisfies G = 1 + zG + tz^2*G + z^3*G^2.

A302483 Number of FF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths.

Original entry on oeis.org

1, 1, 2, 2, 5, 9, 17, 32, 59, 107, 192, 342, 606, 1070, 1885, 3316, 5828, 10237, 17975, 31555, 55387, 97210, 170605, 299405, 525434, 922088, 1618168, 2839704, 4983351, 8745190, 15346758, 26931703, 47261865, 82938813, 145547493, 255418068, 448227487, 786584431

Offset: 0

Sergey Kirgizov, Apr 08 2018

nonn

Number of FF-equivalence classes of Łukasiewicz paths. A Łukasiewicz path of length n is a lattice path from (0,0) to (n,0) using up steps U_{k} = (1,k) for any positive integer k, flat steps F = (1,0) and down steps D = (1,-1). Łukasiewicz paths are alpha-equivalent whenever the positions of occurrences of pattern alpha are identical on these paths.

			There are 14 Łukasiewicz of length 4 divided in the 5 following FF-equivalence classes: {FFFF}, {FFU_{1}D}, {U_{1}DFF}, {U_{1}FFD}, {FU_{1}DF, FU_{1}FD, FU_{2}DD, U_{1}DU_{1}D, U_{1}FDF, U_{1}U_{1}DD, U_{2}DDF, U_{2}DFD, U_{2}FDD, U_{3}DDD}.

Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.

Cf. A001405, A191385, A000045, A005251, A000325, A011782, A001006, A023431, A292460, A004148 enumerates the numbers of P-equivalence classes of Łukasiewicz paths for other values of P.

CoefficientList[Series[(1 - 3 x + 4 x^2 - 5 x^3 + 7 x^4 - 7 x^5 + 6 x^6 - 3 x^7 + x^8)/((1 - 2 x + x^2 - x^3) (1 - x)^2), {x, 0, 32}], x] (* Michael De Vlieger, Apr 12 2018 *)

x='x+O('x^99); Vec((1-3*x+4*x^2-5*x^3+7*x^4-7*x^5+6*x^6-3*x^7+x^8)/((1-2*x+x^2-x^3)*(1-x)^2)) \\ Altug Alkan, Apr 12 2018

G.f.: (1 - 3*x + 4*x^2 - 5*x^3 + 7*x^4 - 7*x^5 + 6*x^6 - 3*x^7 + x^8) / ((1-2*x+x^2-x^3) * (1-x)^2).

cp's OEIS Frontend

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

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A364539 G.f. satisfies A(x) = 1 + x*A(x) + x^3*A(x)^5.

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PARI

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A126218 Triangle read by rows: T(n,k) is the number of 0-1-2 trees (i.e., ordered trees with all vertices of outdegree at most two) with n edges and k pairs of adjacent vertices of outdegree 2.

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Maple

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A166284 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDD's (0<=k<=floor(n/2); U=(1,1), D=(1,-1)).

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Crossrefs

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Maple

Formula

A302483 Number of FF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364539 G.f. satisfies A(x) = 1 + xA(x) + x^3A(x)^5.