|
| |
|
|
A054336
|
|
A convolution triangle of numbers based on A001405 (central binomial coefficients).
|
|
10
| |
|
|
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; 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, June 7, 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. [From Philippe DELEHAM, Dec 08 2009]
Contribution 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))/(2sqrt(1-2x))),
inverse of Riordan array ((1+x)/(1+2x+2x^2),x(1+x)/(1+2x+2x^2)) (A181472). (End)
|
|
|
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, j>=0}A053121(n,j)*binomial(j,k). - Philippe DELEHAM, Mar 30 2007
|
|
|
EXAMPLE
| {1}; {1,1}; {2,2,1}; {3,5,3,1};...
Fourth row polynomial (n=3): p(3,x)= 3+5*x+3*x^2+x^3
Contribution 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]] (* From Jean-François Alcover, Jul 13 2011, after P. Deleham *)
|
|
|
CROSSREFS
| Cf. A001405, A035324, A054335 . Row sums: A054341(n).
Sequence in context: A182810 A139375 A106198 * A079956 A140717 A160232
Adjacent sequences: A054333 A054334 A054335 * A054337 A054338 A054339
|
|
|
KEYWORD
| easy,nice,nonn,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 13 2000
|
| |
|
|
