A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.
1, 2, 1, 5, 4, 1, 15, 14, 6, 1, 51, 50, 27, 8, 1, 188, 187, 113, 44, 10, 1, 731, 730, 468, 212, 65, 12, 1, 2950, 2949, 1956, 970, 355, 90, 14, 1, 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1, 51822, 51821, 35643, 19474, 8612, 3021, 805, 152, 18, 1
Offset: 0
Examples
Triangle begins: 1 2, 1 5, 4, 1 15, 14, 6, 1 51, 50, 27, 8, 1 188, 187, 113, 44, 10, 1 731, 730, 468, 212, 65, 12, 1 2950, 2949, 1956, 970, 355, 90, 14, 1 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1 Production matrix begins 2, 1 1, 2, 1 1, 1, 2, 1 1, 1, 1, 2, 1 1, 1, 1, 1, 2, 1 1, 1, 1, 1, 1, 2, 1 1, 1, 1, 1, 1, 1, 2, 1 ... - _Philippe Deléham_, Mar 01 2013
Links
- Robert Israel, Table of n, a(n) for n = 0..5049
- D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
- D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
Crossrefs
Programs
-
Maple
T := (n,k) -> binomial(n,k)*hypergeom([k/2+1/2,k/2+1,k-n],[k+1,k+2],-4); seq(print(seq(round(evalf(T(n,k),99)),k=0..n)),n=0..8); # Peter Luschny, Sep 23 2014 # Alternative: N:= 12: # to get the first N rows P:= Matrix(N,N,(i,j) -> binomial(i-1,j-1), shape=triangular[lower]): C:= Matrix(N,N,(i,j) -> binomial(2*i-j-1,i-j)*j/i, shape=triangular[lower]): T:= P . C: for i from 1 to N do seq(T[i,j],j=1..i) od; # Robert Israel, Sep 23 2014
-
Mathematica
Flatten[Table[Sum[Binomial[n,i]Binomial[2i-k,i-k](k+1)/(i+1),{i,k,n}],{n,0,100},{k,0,n}]] (* Emanuele Munarini, May 18 2011 *) -
Maxima
create_list(sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i,k,n),n,0,12,k,0,n); /* Emanuele Munarini, May 18 2011 */
Formula
T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
T(n+1,k+2) = T(n+1,k+1) + T(n,k+2) - T(n,k+1) - T(n,k). - Emanuele Munarini, May 18 2011
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i). - Philippe Deléham, Feb 23 2012
T(n,k) = C(n,k)*hypergeom([k/2+1/2,k/2+1,k-n],[k+1,k+2],-4). - Peter Luschny, Sep 23 2014
Comments