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A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients). 13
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, 70, 186, 281, 276, 195, 98, 35, 8, 1, 126, 386, 618, 682, 541, 321, 140, 44, 9, 1, 252, 772, 1362, 1624, 1440, 966, 497, 192, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
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.
Column sequences: A001405, A045621.
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
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
Row sums: A054341.
Sequence in context: A139375 A106198 A202847 * A342908 A284644 A079956
KEYWORD
easy,nice,nonn,tabl
AUTHOR
Wolfdieter Lang, Mar 13 2000
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)