A000344 -id:A000344 - cp's OEIS frontend

A107842 A number triangle of lattice walks.

1, 2, 1, 5, 5, 1, 14, 20, 8, 1, 42, 75, 44, 11, 1, 132, 275, 208, 77, 14, 1, 429, 1001, 910, 440, 119, 17, 1, 1430, 3640, 3808, 2244, 798, 170, 20, 1, 4862, 13260, 15504, 10659, 4655, 1309, 230, 23, 1, 16796, 48450, 62016, 48279, 24794, 8602, 2000, 299, 26, 1

Offset: 0

Paul Barry, May 24 2005

First column is A000108(n+1). Columns include A000344, A003518 and A000589. Row sums are A026671. Compare [1,1,1,...] DELTA [0,1,0,0,...] where DELTA is the operator defined in A084938.

Transposed version in A109450. - Philippe Deléham, Jun 05 2007

			Triangle begins
   1;
   2,  1;
   5,  5,  1;
  14, 20,  8,  1;
  42, 75, 44, 11,  1;
Triangle [1,1,1,1,1,...] DELTA [0,1,0,0,0,0,...] begins:
    1;
    1,   0;
    2,   1,   0;
    5,   5,   1,   0;
   14,  20,   8,   1,   0;
   42,  75,  44,  11,   1,   0;
  132, 275, 208,  77,  14,   1,   0; ...

Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Number triangle T(n, k) = (3k+2)*C(2n+k+1, n-k)/(n+2k+2).

Column k has g.f.: x^k*C(x)^(3k+2) where C(x) is the g.f. of A000108.

A109450 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 8, 20, 14, 0, 1, 11, 44, 75, 42, 0, 1, 14, 77, 208, 275, 132, 0, 1, 17, 119, 440, 910, 1001, 429, 0, 1, 20, 170, 798, 2244, 3808, 3640, 1430, 0, 1, 23, 230, 1309, 4655, 10659, 15504, 13260, 4862, 0, 1, 26, 299, 2000

Offset: 0

Philippe Deléham, Aug 26 2005

Row sums : 1, 1, 3, 11, 43, 173, .... (see A026671).

Transposed version in A107842.

			Triangle begins:
1;
0, 1;
0, 1, 2;
0, 1, 5, 5;
0, 1, 8, 20, 14;
0, 1, 11, 44, 75, 42;
0, 1, 14, 77, 208, 275, 132

Cf. A000108, A000344, A003518, A000589, A107842.

T(0, 0) = 1, T(n, 0) = 0 if n>0, T(n, k) = 0 if k>n, T(n, k) = (3n-3k+2)*binomial(3n-k-1, k-1)/(3n-2k+1).

T(n, n) = A000108(n), Catalan numbers.

A228343 The number of ordered trees with bicolored single edges on the boundary.

Original entry on oeis.org

1, 2, 5, 15, 50, 175, 625, 2251, 8142, 29544, 107538, 392726, 1439204, 5292833, 19533241, 72333107, 268728214, 1001448308, 3742866166, 14026901282, 52701685564, 198481560878, 749170991770, 2833635556670, 10738689128460, 40770816357920, 155056284790340, 590644481896972

Offset: 0

Louis Shapiro, Aug 20 2013

nonn

			When n=3 the five trees contribute as follows: UUUDDD 8; UUDDUD, UDUUDD,UUDUDD 2 each; and UDUDUD just 1.

Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8; preprint, 2014.

Cf. A000108, A228197.

Table[FullSimplify[I*2^n - 5/2*Gamma[3+2*n] * HypergeometricPFQRegularized[{1,3/2+n,2+n},{n,5+n},2]],{n,0,20}] (* Vaclav Kotesovec, Jan 31 2014 *)

x = 'x + O('x^66);
C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
gf = (1+x^2*C^5)/(1-2*x);
Vec(gf) \\ Joerg Arndt, Aug 21 2013

G.f.: (1+x^2*C^5)/(1-2*x) where C is the Catalan number generating function (cf. A000108).
D-finite with recurrence: -(n+3)*(n-2)*a(n) +6*(n^2-2)*a(n-1) -4*n*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 25 2013
a(n) -2*a(n-1) = A000344(n). - R. J. Mathar, Aug 25 2013
a(n) ~ 5 * 2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 31 2014

A268329 Expansion of (1 - sqrt(1 - 4*x))^5/16.

Original entry on oeis.org

2, 10, 40, 150, 550, 2002, 7280, 26520, 96900, 355300, 1307504, 4828850, 17895150, 66533250, 248124000, 927983760, 3479939100, 13082337900, 49295766000, 186156379500, 704415740028, 2670587146260, 10142836030240, 38586876202000, 147029304149000

Offset: 5

Ran Pan, Feb 01 2016

nonn

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal vertically exactly once and horizontally exactly twice, and bounce off the diagonal to the right once but not to the left. Details about this sequence can be found in Section 4.5 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A000344, A000984.

Table[10 Binomial[2 n - 6, n - 5]/n, {n, 5, 29}] (* or *)
Table[SeriesCoefficient[(1 - Sqrt[1 - 4 x])^5/16, {x, 0, n}], {n, 5, 29}] (* Michael De Vlieger, Feb 17 2016 *)

G.f.: (1 - sqrt(1 - 4*x))^5/16.
a(n) = 10 * binomial(2n-6,n-5)/n.
a(n) = 2*A000344(n-3). - R. J. Mathar, Feb 17 2016
D-finite with recurrence: n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Feb 17 2016

cp's OEIS Frontend

A107842 A number triangle of lattice walks.

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A109450 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

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A228343 The number of ordered trees with bicolored single edges on the boundary.

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A268329 Expansion of (1 - sqrt(1 - 4*x))^5/16.

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