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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. 6

%I

%S 1,3,1,11,6,1,45,31,9,1,197,156,60,12,1,903,785,360,98,15,1,4279,3978,

%T 2061,684,145,18,1,20793,20335,11529,4403,1155,201,21,1,103049,104856,

%U 63728,27048,8270,1800,266,24,1,518859,545073,350136,161412,55458,14202

%N 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.

%C This sequence factors A038255 into a product of Riordan arrays.

%H Reinhard Zumkeller, <a href="/A110440/b110440.txt">Rows n = 0..125 of triangle, flattened</a>

%H F. Cai, Q.-H. Hou, Y. Sun, A. L. B. Yang, <a href="http://arxiv.org/abs/1808.05736">Combinatorial identities related to 2x2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018, Table 1.3.

%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

%H Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 7.

%H Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.

%F 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).

%F 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

%F 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

%F Sum_{k, 0<=k<=n} T(n,k)*(2^(k + 1) - 1) = 6^n. - _Philippe Deléham_, Nov 29 2009

%F 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

%F 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

%e Triangle starts:

%e 1;

%e 3,1;

%e 11,6,1;

%e 45,31,9,1;

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

%t 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_ *)

%o (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 */

%o (Maxima)

%o 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 */

%o (Sage)

%o def A110440_triangle(dim):

%o T = matrix(SR,dim,dim)

%o for n in range(dim): T[n,n] = 1

%o for n in (1..dim-1):

%o for k in (0..n-1):

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

%o return T

%o A110440_triangle(9) # _Peter Luschny_, Sep 20 2012

%o (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

%o (Haskell)

%o a110440 n k = a110440_tabl !! n !! k

%o a110440_row n = a110440_tabl !! n

%o a110440_tabl = iterate (\xs ->

%o zipWith (+) ([0] ++ xs) $

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

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

%o -- _Reinhard Zumkeller_, Nov 21 2013

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

%K easy,nice,nonn,tabl

%O 0,2

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

%E Typo in recursion formula fixed by _Reinhard Zumkeller_, Nov 21 2013

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)