login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A110440 Triangular array formed by the little Schroeder numbers. s(n,k)= the number of unit step restricted paths (i.e., they never go below the x-axis) from the origin (0,0) to (n-1,k-1) using up step U(1,1), three types of level steps L(1,0), L'(1,0), L"(1,0) and two types of down steps D(1,-1), D'(1,-1). s(0,0)=1 and the leftmost column s(n,0) is A001003. 5
1, 3, 1, 11, 6, 1, 45, 31, 9, 1, 197, 156, 60, 12, 1, 903, 785, 360, 98, 15, 1, 4279, 3978, 2061, 684, 145, 18, 1, 20793, 20335, 11529, 4403, 1155, 201, 21, 1, 103049, 104856, 63728, 27048, 8270, 1800, 266, 24, 1, 518859, 545073, 350136, 161412, 55458, 14202 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence factors A038255 into a product of Riordan arrays.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

F. Cai, Q.-H. Hou, Y. Sun, A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018, Table 1.3.

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 7.

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

FORMULA

Recurrence is s(n+1,0) = 3s(n,0) + 2s(n,1) and for k > 0: s(n+1,k) = s(n,k-1) + 3s(n,k) + 2s(n,k+1). Riordan array ((1 - 3z - sqrt(1-6z+z^2))/4z*z, (1 - 3z - sqrt(1-6z+z^2))/4z).

Sum_{k, k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n+1). - Philippe Deléham, Sep 14 2005

G.f.: 2/( 1 - x*L -2*x*y*U + sqrt( (1 - x*L)^2 - 4*x^2*D*U ) ) where L=3, U = 1, D = 2. - Michael Somos, Mar 31 2007

Sum_{k, 0<=k<=n} T(n,k)*(2^(k + 1) - 1) = 6^n. - Philippe Deléham, Nov 29 2009

T(n,k) = sum(i = 0..k + 1, (i*(-1)^(k - i + 1)*binomial(k + 1, i)*sum(j = 0..n + 1, (-1)^j*2^(n + 1 - j)*(2*n + i - j + 1)!/((n + i - j + 1)!*j!*(n - j + 1)!)))). - Vladimir Kruchinin, Oct 17 2011

T(n,k) = ((k + 1)/(n + 1)*sum(j = ceiling((n + k + 2)/2)..n + 1, binomial(j, -n - k + 2*j - 2)*3^(-n - k + 2*j - 2)*2^(n + 1 - j)*binomial(n + 1, j))). - Vladimir Kruchinin, Jan 28 2013

EXAMPLE

Triangle starts:

1;

3,1;

11,6,1;

45,31,9,1;

197,156,60,12,1; ...

MATHEMATICA

nmax = 9; t[n_, k_] := Sum[(i*(-1)^(k-i+1)*Binomial[k+1, i]*Sum[(-1)^j*2^(n+1-j)*(2n+i-j+1)! / ((n+i-j+1)!*j!*(n-j+1)!), {j, 0, n+1}]), {i, 0, k+1}]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)

PROG

(PARI) {T(n, k)= if(n<0| k>n, 0, polcoeff(polcoeff( 2/(1 -3*x -2*x*y +sqrt( 1 -6*x +x^2 +x*O(x^n)) ), n), k))} /* Michael Somos, Mar 31 2007 */

(Maxima)

T(n, k):=sum((i*(-1)^(k-i+1)*binomial(k+1, i)*sum((-1)^j*2^(n+1-j)*(2*n+i-j+1)!/((n+i-j+1)!*j!*(n-j+1)!), j, 0, n+1)), i, 0, k+1); /* Vladimir Kruchinin, Oct 17 2011 */

(Sage)

def A110440_triangle(dim):

    T = matrix(SR, dim, dim)

    for n in range(dim): T[n, n] = 1

    for n in (1..dim-1):

        for k in (0..n-1):

            T[n, k] = T[n-1, k-1]+3*T[n-1, k]+2*T[n-1, k+1]

    return T

A110440_triangle(9) # Peter Luschny, Sep 20 2012

(Maxima) T(n, k):=((k+1)/(n+1)*sum(binomial(j, -n-k+2*j-2)*3^(-n-k+2*j-2)*2^(n+1-j)*binomial(n+1, j), j, ceiling((n+k+2)/2), n+1)); \\ Vladimir Kruchinin, Jan 28 2013

(Haskell)

a110440 n k = a110440_tabl !! n !! k

a110440_row n = a110440_tabl !! n

a110440_tabl = iterate (\xs ->

   zipWith (+) ([0] ++ xs) $

   zipWith (+) (map (* 3) (xs ++ [0]))

               (map (* 2) (tail xs ++ [0, 0]))) [1]

-- Reinhard Zumkeller, Nov 21 2013

CROSSREFS

Cf. A232246 (central terms), A001003 (left column), A065096 (2nd column?), A225887 (row sums?).

Sequence in context: A113955 A110165 A111965 * A135574 A008969 A199577

Adjacent sequences:  A110437 A110438 A110439 * A110441 A110442 A110443

KEYWORD

easy,nice,nonn,tabl

AUTHOR

Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005

EXTENSIONS

Typo in recursion formula fixed by Reinhard Zumkeller, Nov 21 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 19:18 EDT 2018. Contains 316293 sequences. (Running on oeis4.)