A110320 -id:A110320 - cp's OEIS frontend

A246182 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k hh's. 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 h of weight 1; a (1,0)-step H of weight 2; a (1,1)-step u of weight 2; a (1,-1)-step d of weight 1. The weight of a path is the sum of the weights of its steps.

1, 1, 1, 1, 3, 0, 1, 5, 2, 0, 1, 9, 5, 2, 0, 1, 19, 9, 6, 2, 0, 1, 39, 21, 12, 7, 2, 0, 1, 79, 53, 27, 15, 8, 2, 0, 1, 167, 118, 74, 34, 18, 9, 2, 0, 1, 357, 269, 180, 96, 42, 21, 10, 2, 0, 1, 763, 639, 419, 254, 119, 51, 24, 11, 2, 0, 1, 1651, 1486, 1045, 605, 340, 143, 61, 27, 12, 2, 0, 1

Offset: 0

Emeric Deutsch, Aug 23 2014

Number of entries in row n is n (n>=1).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
Sum(k*T(n,k), k>=0) = A110320(n-1) (n>=1).

			Row 3 is 3,0,1. Indeed, the four paths of weight 3 are: ud, hH, Hh, and hhh, having 0, 0, 0, and 2 hh's, respectively.
Triangle starts:
1;
1;
1,1;
3,0,1;
5,2,0,1;
9,5,2,0,1;

Alois P. Heinz, Rows n = 0..141, flattened
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Cf. A004148, A110320, A246177.

eq := z^3*(1+z-t*z)*G^2-(-z^3+1-z^2-t*z+t*z^3)*G+1+z-t*z = 0: g := RootOf(eq, G): gser := simplify(series(g, z = 0, 18)): for j from 0 to 15 do P[j] := coeff(gser, z, j) end do: 1; for j to 13 do seq(coeff(P[j], t, q), q = 0 .. j-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
      expand(b(n-1, y, 1)*`if`(t=1, x, 1)+ `if`(n>1, b(n-2, y, 0)+
      b(n-2, y+1, 0), 0) +b(n-1, y-1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Aug 24 2014

b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y, 1] * If[t==1, x, 1] + If[n>1, b[n-2, y, 0] + b[n-2, y+1, 0], 0] + b[n-1, y-1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

G.f. G=G(t,z) satisfies z^3*(1+z-t*z)*G^2 - (1-t*z-z^2+t*z^3-z^3)*G+1+z-t*z=0.

A246183 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k HH's. 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 h of weight 1; a (1,0)-step H of weight 2; a (1,1)-step u of weight 2; a (1,-1)-step d of weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 4, 7, 1, 15, 2, 33, 3, 1, 71, 9, 2, 158, 23, 3, 1, 357, 54, 10, 2, 812, 136, 26, 3, 1, 1869, 338, 63, 11, 2, 4338, 835, 167, 29, 3, 1, 10134, 2087, 428, 72, 12, 2, 23829, 5216, 1092, 199, 32, 3, 1, 56341, 13046, 2826, 523, 81, 13, 2

Offset: 0

Emeric Deutsch, Aug 23 2014

Number of entries in row n is floor(n/2) (n>=2).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
Sum(k*T(n,k), k>=0) = A110320(n-3) (n>=4).

			Row 3 is 4. Indeed, the four paths of weight 3 are: ud, hH, Hh, and hhh; none of them contain HH.
Triangle starts:
1;
1;
2;
4;
7,1;
15,2;
33,3,1;

Alois P. Heinz, Rows n = 0..200, flattened
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Cf. A004148, A110320.

eq := z^3*(1+z^2-t*z^2)*G^2-(1-z-t*z^2+t*z^3-z^3)*G+1+z^2-t*z^2 = 0: g := RootOf(eq, G): gser := simplify(series(g, z = 0, 22)): for j from 0 to 20 do P[j] := sort(coeff(gser, z, j)) end do: 1; for j to 20 do seq(coeff(P[j], t, q), q = 0 .. (1/2)*j-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
      expand(b(n-1, y, 0)+ `if`(n>1, b(n-2, y, 1)*`if`(t=1, x, 1)+
      b(n-2, y+1, 0), 0) +b(n-1, y-1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..20); # Alois P. Heinz, Aug 24 2014

b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y, 0] + If[n>1, b[n-2, y, 1]*If[t==1, x, 1] + b[n-2, y+1, 0], 0] + b[n-1, y-1, 0]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

G.f. G=G(t,z) satisfies z^3*(1+z^2-t*z^2)*G^2 - (1-z-t*z^2+t*z^3-z^3)*G +1+z^2-t*z^2=0.

A247297 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uudd strings.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 36, 1, 80, 2, 180, 5, 410, 13, 946, 32, 2203, 80, 5173, 199, 1, 12233, 499, 3, 29108, 1255, 9, 69643, 3161, 28, 167437, 7984, 81, 404311, 20206, 231, 980125, 51228, 650, 1, 2384441, 130090, 1812, 4, 5819576, 330835, 5016, 14

Offset: 0

Emeric Deutsch, Sep 17 2014

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: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
Row n contains 1 + floor(n/6) entries.
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A247298(n).
Sum(k*T(n,k), k=0..n) =A110320(n-5) (n>=6)

			T(6,1)=1 because among the 37 (=A004148(7)) paths in B(6) only uudd contains uudd.
T(13,2)=3 because we have huudduudd, uuddhuudd, and uudduuddh.
Triangle starts:
1;
1;
2;
4;
8;
17;
36,1;
80,2;

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

Cf. A004148, A110320, A247298.

eq := G = 1+z*G+z^2*G+z^3*(G-z^3+t*z^3)*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, k), k = 0 .. floor((1/6)*n)) end do; # yields sequence in triangular form

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

A140639 Number of 4-noncrossing RNA structures with arc-length => 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 15, 51, 179, 647, 2397, 9081, 35181, 139307, 563218

Offset: 1

Jonathan Vos Post, Jul 07 2008

nonn

Table 2, p. 30 of Han and Reidys. See diagrams p. 31. Combinatorial formulas and asymptotic results are also given.

Hillary S. W. Han, Christian M. Reidys, Pseudoknot RNA structures with arc-length => 4

Cf. A004148, A110319, A107905, A110320, A132441.

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.

A298309 Triangle read by rows: T(n,m) = Sum_{i=0..n+1} C(n-i+1,i-1)C(n-i+1,i)C(n-i+1,m-i+1).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 11, 13, 5, 4, 25, 51, 43, 13, 5, 49, 149, 203, 130, 32, 6, 86, 364, 716, 734, 382, 80, 7, 139, 787, 2099, 3061, 2521, 1105, 201, 8, 211, 1553, 5385, 10455, 12093, 8311, 3143, 505, 9, 305, 2851, 12473, 30918, 47064, 45075, 26581, 8843, 1273

Offset: 0

Vladimir Kruchinin, Jan 17 2018

			Triangle begins
  0;
  1,    1;
  2,    4,    2;
  3,   11,   13,    5;
  4,   25,   51,   43,   13;
  5,   49,  149,  203,  130,   32;
  6,   86,  364,  716,  734,  382,   80;
  7,  139,  787, 2099, 3061, 2521, 1105,  201;

T(n,n) is A110320(n).

T(n,m):=sum(binomial(n-i+1,i-1)*binomial(n-i+1,i)*binomial(n-i+1,m-i+1),i,0,n+1);

T(n,m) = sum(i=0, n+1, binomial(n-i+1,i-1)*binomial(n-i+1,i)*binomial(n-i+1,m-i+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Jan 19 2018

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

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A247297 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uudd strings.

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A140639 Number of 4-noncrossing RNA structures with arc-length => 4.

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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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A298309 Triangle read by rows: T(n,m) = Sum_{i=0..n+1} C(n-i+1,i-1)*C(n-i+1,i)*C(n-i+1,m-i+1).

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A298309 Triangle read by rows: T(n,m) = Sum_{i=0..n+1} C(n-i+1,i-1)C(n-i+1,i)C(n-i+1,m-i+1).