|
| |
|
|
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.
|
|
2
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| This sequence factors A038255 into a product of Riordan arrays.
|
|
|
REFERENCES
| 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.
|
|
|
FORMULA
| Recurrence is s(n+1, 0)= 3s(n, k)+ 2s(n, k+1) for the leftmost column entries and s(n+1, k)= s(n, k-1)+ 3s(n, k)+ 2s(n, k+1) for the other column entries. 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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)!)))). [From Vladimir Kruchinin, Oct 17 2011]
|
|
|
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}]] (* From 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); [From Vladimir Kruchinin, Oct 17 2011]
|
|
|
CROSSREFS
| 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
|
| |
|
|
