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%I #36 Aug 13 2020 12:13:15
%S 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 212,65,12,1,2950,2949,1956,970,355,90,14,1,12235,12234,8291,4356,
%U 1785,550,119,16,1,51822,51821,35643,19474,8612,3021,805,152,18,1
%N Triangle T read by rows: matrix product of Pascal and Catalan triangle.
%C Also, Riordan array (G,G), G(t)=(1 - ((1-5*t)/(1-t))^(1/2))/(2*t).
%C From _Emanuele Munarini_, May 18 2011: (Start)
%C Row sums = A002212.
%C Diagonal sums = A190737.
%C Central coefficients = A190738. (End)
%H Robert Israel, <a href="/A104259/b104259.txt">Table of n, a(n) for n = 0..5049</a>
%H 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 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 T(n,k) = sum(binomial(n,i)*binomial(2*i-k,i-k)*(k+1)/(i+1),i=k..n).
%F 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 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 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 Triangle begins:
%e 1
%e 2, 1
%e 5, 4, 1
%e 15, 14, 6, 1
%e 51, 50, 27, 8, 1
%e 188, 187, 113, 44, 10, 1
%e 731, 730, 468, 212, 65, 12, 1
%e 2950, 2949, 1956, 970, 355, 90, 14, 1
%e 12235, 12234, 8291, 4356, 1785, 550, 119, 16, 1
%e Production matrix begins
%e 2, 1
%e 1, 2, 1
%e 1, 1, 2, 1
%e 1, 1, 1, 2, 1
%e 1, 1, 1, 1, 2, 1
%e 1, 1, 1, 1, 1, 2, 1
%e 1, 1, 1, 1, 1, 1, 2, 1
%e ... - _Philippe Deléham_, Mar 01 2013
%p 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 # Alternative:
%p N:= 12: # to get the first N rows
%p P:= Matrix(N,N,(i,j) -> binomial(i-1,j-1), shape=triangular[lower]):
%p C:= Matrix(N,N,(i,j) -> binomial(2*i-j-1,i-j)*j/i, shape=triangular[lower]):
%p T:= P . C:
%p for i from 1 to N do
%p seq(T[i,j],j=1..i)
%p od; # _Robert Israel_, Sep 23 2014
%t 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 (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 T = A007318 * A033184.
%Y Left-hand columns include A007317, A007317 - 1. Row sums are in A002212.
%K nonn,tabl
%O 0,2
%A _Ralf Stephan_, Mar 17 2005
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