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A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.

%I A104259 #36 Aug 13 2020 12:13:15
%S A104259 1,2,1,5,4,1,15,14,6,1,51,50,27,8,1,188,187,113,44,10,1,731,730,468,
%T A104259 212,65,12,1,2950,2949,1956,970,355,90,14,1,12235,12234,8291,4356,
%U A104259 1785,550,119,16,1,51822,51821,35643,19474,8612,3021,805,152,18,1
%N A104259 Triangle T read by rows: matrix product of Pascal and Catalan triangle.
%C A104259 Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).
%C A104259 From _Emanuele Munarini_, May 18 2011: (Start)
%C A104259 Row sums = A002212.
%C A104259 Diagonal sums = A190737.
%C A104259 Central coefficients = A190738. (End)
%H A104259 Robert Israel, <a href="/A104259/b104259.txt">Table of n, a(n) for n = 0..5049</a>
%H A104259 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="https://doi.org/10.1007/978-3-0348-8405-1_11">An algebra for proper generating trees</a>, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [<a href="https://www.researchgate.net/publication/2386023_An_Algebra_for_Proper_Generating_Trees">alternative link</a>]
%H A104259 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://local.disia.unifi.it/merlini/papers/Lucidi.ps">An algebra for proper generating trees</a>, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
%F A104259 T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
%F A104259 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
%F A104259 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
%F A104259 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
%e A104259 Triangle begins:
%e A104259 1
%e A104259 2, 1
%e A104259 5, 4, 1
%e A104259 15, 14, 6, 1
%e A104259 51, 50, 27, 8, 1
%e A104259 188, 187, 113, 44, 10, 1
%e A104259 731, 730, 468, 212, 65, 12, 1
%e A104259 2950, 2949, 1956, 970, 355, 90, 14, 1
%e A104259 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1
%e A104259 Production matrix begins
%e A104259 2, 1
%e A104259 1, 2, 1
%e A104259 1, 1, 2, 1
%e A104259 1, 1, 1, 2, 1
%e A104259 1, 1, 1, 1, 2, 1
%e A104259 1, 1, 1, 1, 1, 2, 1
%e A104259 1, 1, 1, 1, 1, 1, 2, 1
%e A104259 ... - _Philippe Deléham_, Mar 01 2013
%p A104259 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
%p A104259 # Alternative:
%p A104259 N:= 12:  # to get the first N rows
%p A104259 P:= Matrix(N,N,(i,j) -> binomial(i-1,j-1), shape=triangular[lower]):
%p A104259 C:= Matrix(N,N,(i,j) -> binomial(2*i-j-1,i-j)*j/i, shape=triangular[lower]):
%p A104259 T:= P . C:
%p A104259 for i from 1 to N do
%p A104259 seq(T[i,j],j=1..i)
%p A104259 od;   # _Robert Israel_, Sep 23 2014
%t A104259 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 *)
%o A104259 (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 */
%Y A104259 T = A007318 * A033184.
%Y A104259 Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.
%K A104259 nonn,tabl
%O A104259 0,2
%A A104259 _Ralf Stephan_, Mar 17 2005