A069720 -id:A069720 - cp's OEIS frontend

A103945 Number of rooted dual-unicursal n-edge maps in the plane (planar with a distinguished outside face).

2, 14, 107, 844, 6757, 54522, 441863, 3589880, 29206025, 237780982, 1936486411, 15771410420, 128431734797, 1045618229234, 8510270668815, 69241255165936, 563154350637073, 4578526894227438, 37209886138826771, 302291556342169580

Offset: 1

Valery A. Liskovets, Mar 17 2005

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A069720, A103944.

A069720[n_] := 2^(n-1) Binomial[2n-1, n];
A103944[n_] := If[n == 1, 1, n Binomial[2n, n] Sum[Binomial[n-2, k] (1/(n + 1 + k) + n/(n + 2 + k)), {k, 0, n-2}]];
a[n_] := (n+2) A069720[n] - A103944[n];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

a(n)=(n+2)*A069720(n)-A103944(n).

A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 45, 96, 66, 16, 1, 197, 501, 450, 170, 25, 1, 903, 2668, 2955, 1520, 365, 36, 1, 4279, 14407, 18963, 12355, 4165, 693, 49, 1, 20793, 78592, 119812, 94528, 41230, 9856, 1204, 64, 1, 103049, 432073, 748548, 693588, 372078, 117054

Offset: 0

Emeric Deutsch, Dec 15 2005

Row sums yield A052701. Column 0 yields the little Schroeder numbers (A001003). Sum_{k=0..n} k*T(n,k) = A069720(n).

Triangle T(n,k), 0 <= k <= n, read by rows; given by [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 23 2005

			T(3,2)=9 because we have (ud)(ud)Ud, (ud)Ud(ud), Ud(ud)(ud), (ud)u(ud)d,
(ud)U(ud)d, u(ud)d(ud), U(ud)d(ud), u(ud)(ud)d and U(ud)(ud)d (the ud peaks are shown between parentheses).
Triangle starts:
   1;
   1,  1;
   3,  4,  1;
  11, 19,  9,  1;
  45, 96, 66, 16,  1;

Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

Cf. A001003, A052701, A069720.

T:=proc(n,k) if k<=n-1 then (1/n)*binomial(n,k)*sum(2^j*binomial(n,j+1)*binomial(n-k,j),j=0..n-k) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] := If[k <= n-1, (1/n)*Binomial[n, k]*Sum[2^j*Binomial[n, j+1]* Binomial[n-k, j], {j, 0, n-k}], If[k == n, 1, 0]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2018, from Maple *)

T(n,k) = (1/n)*binomial(n,k)*Sum_{j=0..n-k} 2^j*binomial(n, j+1)*binomial(n-k, j) (k <= n-1); T(n, n)=1.

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

A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

Original entry on oeis.org

1, 6, 1, 40, 10, 1, 280, 84, 14, 1, 2016, 672, 144, 18, 1, 14784, 5280, 1320, 220, 22, 1, 109824, 41184, 11440, 2288, 312, 26, 1, 823680, 320320, 96096, 21840, 3640, 420, 30, 1, 6223360, 2489344, 792064, 198016, 38080, 5440, 544, 34, 1, 47297536, 19348992, 6449664, 1736448, 372096, 62016, 7752

Offset: 0

N. J. A. Sloane, Oct 19 2017

			Triangle begins:
1,
6,1,
40,10,1,
280,84,14,1,
2016,672,144,18,1,
14784,5280,1320,220,22,1,
...

Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091. See p. 3088.

First column is A069720.

A293172 := proc(n,k)
    option remember;
    local b,d,r,c,e;
    b := 4; d:= 2; r := 2 ; c := r^2 ; e := d ;
    if k < 0 or k > n then
        0;
    elif k = n then
        1;
    elif k = 0 then
        (b+e)*procname(n-1,0)+c*procname(n-1,1) ;
    else
        procname(n-1,k-1)+b*procname(n-1,k)+c*procname(n-1,k+1) ;
    end if;
end proc:
seq(seq( A293172(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Oct 27 2017

T[n_, k_] := T[n, k] = Module[{b = 4, d = 2, r = 2, c, e}, c = r^2; e = d; If[k < 0 || k > n, 0, If[k == n, 1, If[k == 0, (b + e) T[n - 1, 0] + c T[n - 1, 1], T[n - 1, k - 1] + b T[n - 1, k] + c T[n - 1, k + 1]]]]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 07 2020, from Maple *)

A337977 Triangle T(n,m) = C(n-1,n-m)Sum_{k=1..n} C(2k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 8, 5, 1, 10, 22, 26, 14, 1, 15, 50, 85, 90, 42, 1, 21, 100, 225, 348, 322, 132, 1, 28, 182, 525, 1050, 1442, 1176, 429, 1, 36, 308, 1120, 2730, 4928, 5992, 4356, 1430, 1, 45, 492, 2226, 6426, 14238, 22920, 24894, 16302, 4862

Offset: 1

Vladimir Kruchinin, Oct 05 2020

			1,
1, 1,
1, 3,  2,
1, 6,  8,  5,
1,10, 22, 26, 14,
1,15, 50, 85, 90, 42,
1,21,100,225,348,322,132

T(2*n,n) is A069720.

2nd column: A000217, 3rd column: 2*A006522 or 2*(A027927-1).

Table[Binomial[n - 1, n - m] Sum[Binomial[2 k - 2, k - 1] Binomial[n - m, m - k]/m, {k, n}], {n, 10}, {m, n}] // Flatten (* Michael De Vlieger, Oct 05 2020 *)

T(n,m):=(binomial(n-1,n-m)*sum(binomial(2*k-2,k-1)*binomial(n-m,m-k),k,1,n))/m;

G.f.: A(x,y) = -(sqrt((2*sqrt(-4*x^2*y+x^2-2*x+1)+3*x-2)/(4*x))-1/2).

A360651 Triangle T(n, m) = (n - m + 1)C(2n + 1, m)C(2n - m + 2, n - m + 1)/(2*n - m + 2).

Original entry on oeis.org

1, 3, 3, 10, 20, 10, 35, 105, 105, 35, 126, 504, 756, 504, 126, 462, 2310, 4620, 4620, 2310, 462, 1716, 10296, 25740, 34320, 25740, 10296, 1716, 6435, 45045, 135135, 225225, 225225, 135135, 45045, 6435, 24310, 194480, 680680, 1361360, 1701700, 1361360, 680680, 194480, 24310

Offset: 0

Vladimir Kruchinin, Feb 15 2023

			Triangle T(n, m) starts:
[0] 1;
[1] 3,     3;
[2] 10,    20,    10;
[3] 35,    105,   105,     35;
[4] 126,   504,   756,     504,     126;
[5] 462,   2310,  4620,    4620,    2310,    462;
[6] 1716,  10296, 25740,   34320,   25740,   10296,    1716;
[7] 6435,  45045, 135135,  225225,  225225,  135135,   45045,  6435;

Cf. A001700, A085880, A069720 (row sums).

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
T := (n, k) -> (2*n + 1)*CatalanNumber(n)*binomial(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Feb 15 2023

Maxima
```
T(n,m):=if n
				
```

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

T(n, k) = binomial(n, k)*CatalanNumber(n)*(2*n + 1). - Peter Luschny, Feb 15 2023

cp's OEIS Frontend

A103945 Number of rooted dual-unicursal n-edge maps in the plane (planar with a distinguished outside face).

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A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U.

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A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

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A337977 Triangle T(n,m) = C(n-1,n-m)*Sum_{k=1..n} C(2*k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

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A360651 Triangle T(n, m) = (n - m + 1)*C(2*n + 1, m)*C(2*n - m + 2, n - m + 1)/(2*n - m + 2).

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Formula

A337977 Triangle T(n,m) = C(n-1,n-m)Sum_{k=1..n} C(2k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.

A360651 Triangle T(n, m) = (n - m + 1)C(2n + 1, m)C(2n - m + 2, n - m + 1)/(2*n - m + 2).