A003518 -id:A003518 - cp's OEIS frontend

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4binomial(n+1, k+3)binomial(n+1, k-1)/(n+1).

1, 4, 4, 10, 24, 10, 20, 84, 84, 20, 35, 224, 392, 224, 35, 56, 504, 1344, 1344, 504, 56, 84, 1008, 3780, 5760, 3780, 1008, 84, 120, 1848, 9240, 19800, 19800, 9240, 1848, 120, 165, 3168, 20328, 58080, 81675, 58080, 20328, 3168, 165, 220, 5148, 41184, 151008

Offset: 3

Peter Bala, Oct 15 2008

T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 3 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.

The current array is the case r = 3 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145599 (r = 4).

			Triangle starts
  n\k|  1     2     3     4     5     6
  =====================================
   3 |  1
   4 |  4     4
   5 | 10    24    10
   6 | 20    84    84    20
   7 | 35   224   392   224    35
   8 | 56   504  1344  1344   504    56
  ...
Row 5: T(5,3) = 10: the 10 walks of length 5 from (0,0) to (2,3) are UUURR, UURUR, UURRU, URUUR, URURU, URRUU, RUUUR, RUURU, RURUU and RRUUU.
*
*......*......*......y......*......*......*
.
.
*.....10......*.....24......*.....10......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......o......*......*......* x axis
.

F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=3.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Cf. A003518 (row sums), A001263, A145602, A145596, A145597, A145599.

T := (n,k) -> 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1):
for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do;

T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1) for n >=3 and 1 <= k <= n-2. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 1,3). Row sums A003518.
O.g.f. for column k+2: 4/(k + 1) * y^(k+4)/(1 - y)^(k+6) * Jacobi_P(k,4,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) = Sum_{k = 1 .. n - 2} T(n,k)*x^k:
x^3*R_(n-1)(x) = 4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)*(n + 4)) * Sum_{k = 0..n} binomial(n + 4,k) * binomial(2n - k,n) * (x - 1)^k;
Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-1) = A003518(n)*x^(n-1).
Row generating polynomial R_(n+3)(x) = 4/(n+4)*x*(1-x)^n * Jacobi_P(n,4,4,(1+x)/(1-x)). - Peter Bala, Oct 31 2008
G.f.: A(x) = x*A145596(x)^2. - Vladimir Kruchinin, Oct 09 2020

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

A026018 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n-1) = 7. Also a(n) = T(2n-1,n-3), where T is the array defined in A026009.

Original entry on oeis.org

1, 7, 36, 164, 702, 2898, 11696, 46512, 183141, 716243, 2788060, 10817820, 41880930, 161900910, 625272480, 2413491360, 9313307370, 35936613414, 138680365704, 535290282632, 2066802226236, 7983111461732, 30848211650592

Offset: 3

Clark Kimberling

nonn

First differences if A003518.

Conjecture: -(n+5)*(3*n-37)*a(n) +3*(-n^2-84*n-173)*a(n-1) +2*(32*n^2+295*n+254)*a(n-2) -8*(n+25)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jun 20 2013

A107842 A number triangle of lattice walks.

Original entry on oeis.org

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.

cp's OEIS Frontend

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4*binomial(n+1, k+3)*binomial(n+1, k-1)/(n+1).

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A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

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A026018 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n-1) = 7. Also a(n) = T(2n-1,n-3), where T is the array defined in A026009.

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A107842 A number triangle of lattice walks.

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Formula

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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Comments

Examples

Crossrefs

Formula

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4binomial(n+1, k+3)binomial(n+1, k-1)/(n+1).