OFFSET
0,4
COMMENTS
T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height and having k blue level steps. Example: T(4,2)=9 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BBRR, BRBR, BRRB, RBBR, RBRB, RRBB, BBUD, BUDB, and UDBB. - Emeric Deutsch, Jun 07 2011
In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(1-(1+x)*z-z^2*c(z^2)), with c(x) the g.f. for Catalan numbers A000108.
Riordan array (f(x), x*f(x)), f(x) the g.f. of A001405. - Philippe Deléham, Dec 08 2009
From Paul Barry, Oct 21 2010: (Start)
Riordan array ((sqrt(1+2x) - sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2*sqrt(1-2x))),
inverse of Riordan array ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2)) (A181472). (End)
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
G.f. for column m: cbi(x)*(x*cbi(x))^m, with cbi(x) := (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)), where c(x) is the g.f. for Catalan numbers A000108.
T(n,k) = Sum_{j>=0} A053121(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
T(n,k) = T(n-1,k-1) + T(n-1,l) + Sum_{j>=0} T(n-1,k+1+j)*(-1)^j. - Philippe Deléham, Feb 23 2012
EXAMPLE
Fourth row polynomial (n=3): p(3,x)= 3 + 5*x + 3*x^2 + x^3.
From Paul Barry, Oct 21 2010: (Start)
Triangle begins
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
6, 10, 9, 4, 1;
10, 22, 22, 14, 5, 1;
20, 44, 54, 40, 20, 6, 1;
35, 93, 123, 109, 65, 27, 7, 1;
Production matrix is
1, 1;
1, 1, 1;
-1, 1, 1, 1;
1, -1, 1, 1, 1;
-1, 1, -1, 1, 1, 1;
1, -1, 1, -1, 1, 1, 1;
-1, 1, -1, 1, -1, 1, 1, 1;
1, -1, 1, -1, 1, -1, 1, 1, 1;
-1, 1, -1, 1, -1, 1, -1, 1, 1, 1; (End)
MATHEMATICA
c[n_, j_] /; n < j || OddQ[n - j] = 0; c[n_, j_] = (j + 1) Binomial[n + 1, (n - j)/2]/(n + 1); t[n_, k_] := Sum[c[n, j]*Binomial[j, k], {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[;; 66]] (* Jean-François Alcover, Jul 13 2011, after Philippe Deléham *)
PROG
(PARI)
A053121(n, k) = if((n-k+1)%2==0, 0, (k+1)*binomial(n+1, (n-k)\2)/(n+1) );
T(n, k) = sum(j=k, n, A053121(n, j)*binomial(j, k));
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 21 2019
(Magma)
A053121:= func< n, k | ((n-k+1) mod 2) eq 0 select 0 else (k+1)*Binomial(n+1, Floor((n-k)/2))/(n+1) >;
T:= func< n, k | (&+[Binomial(j, k)*A053121(n, j): j in [k..n]]) >;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 21 2019
(Sage)
def A053121(n, k):
if (n-k+1) % 2==0: return 0
else: return (k+1)*binomial(n+1, ((n-k)//2))/(n+1)
def T(n, k): return sum(binomial(j, k)*A053121(n, j) for j in (k..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 21 2019
(GAP)
A053121:= function(n, k)
if ((n-k+1) mod 2)=0 then return 0;
else return (k+1)*Binomial(n+1, Int((n-k)/2))/(n+1);
fi;
end;
T:= function(n, k)
return Sum([k..n], j-> Binomial(j, k)*A053121(n, j));
end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 21 2019
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 13 2000
STATUS
approved