A110440 Triangular array formed by the little Schröder numbers s(n,k).
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
Offset: 0
Examples
Triangle starts: [0] 1; [1] 3, 1; [2] 11, 6, 1; [3] 45, 31, 9, 1; [4] 197, 156, 60, 12, 1; [5] 903, 785, 360, 98, 15, 1; [6] 4279, 3978, 2061, 684, 145, 18, 1; [7] 20793, 20335, 11529, 4403, 1155, 201, 21, 1; [8] 103049, 104856, 63728, 27048, 8270, 1800, 266, 24, 1; [9] 518859, 545073, 350136, 161412, 55458, 14202, 2646, 340, 27, 1;
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- 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 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.
Crossrefs
Programs
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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 -
Maple
T := (n, k) -> ((k + 1)/(n + 1))*add(2^(n - m)*binomial(n+1, m+1)*binomial(n+1, m-k), m = 0..n): seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jan 09 2022
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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 *) -
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 */
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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 */
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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 -
Sage
def A110440_triangle(dim): T = matrix(ZZ,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
Formula
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). [Typo fixed by Reinhard Zumkeller, Nov 21 2013]
Riordan array ((1 - 3z - sqrt(1-6z+z^2))/4z*z, (1 - 3z - sqrt(1-6z+z^2))/4z).
Sum_{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..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)*C(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} C(j,2*j-n-k-2)*3^(2*j-n-k- 2)*2^(n+1-j)*C(n+1,j). - Vladimir Kruchinin, Jan 28 2013
T(n,k) = ((k+1)/(n+1))*Sum_{m=0..n} 2^(n-m)*C(n+1,m+1)*C(n+1,m-k). - Vladimir Kruchinin, Jan 09 2022
Comments